101 mid1 - Bogazici University Physics Department Phys 101 Midterm 1 H. Beker Fall '97 Question 3: the potential energy function on a plane is given by U(x, y) = −2x3 y2 (MKS units)

1

Time 50 min Phys 101 Midterm 1 H. Beker Fall '80

No : . Name : . Total Grade : . Grade : .

Question 1 : Find a vector of length 15 and perpendicular to both 2ˆ Â i j= += += += +�

and 2ˆ ˆB i k= += += += +�

.

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Phys 101 Midterm 1 H. Beker

Question 1 : Find a unit vector that is perpendicular to both 4ˆ ˆ Â i j k= − += − += − += − +�

and ˆ ˆ ˆB i j k= + −= + −= + −= + −�

.

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Phys 101 Midterm 1

Question 4 : Find the T for : a) 0=µ b) 2.0=µ c) 4.0=µ .

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Phys 101 Midterm 1 ?? ??

Question 4 : A force F pushes M such that m is stationary with respect to M :

a) Draw the free body diagram for m

b) Draw the free body diagram for M

c) Find the friction between M and m

d) Find the friction between M and the table

e) Find the minimum necessary force F

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Phys 101 Midterm 1

Question 4 : The position of a 3 kg object is given by jtittr ˆ2ˆ)( 2 −=�

(in MKS units) . (a) Find the kinetic

energy at 1=t sec , (b) Find the kinetic energy at 2=t sec , (c) Find the work done in that time interval .

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Phys 101 Midterm 1 F. Neyzi

Question 4 : A model airplane of mass M flies in a horizontal circle with constant

speed 0v at the end of a string of length l . The string makes a constant angle θ with the

horizontal. The aerodynamic lift acts perpendicular to the string as shown. Find the

tension T in the string in terms of M , 0v , l , θ , g.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 Question 4 : A particle moving in 1-D with constant acceleration passes two consecutive equal space intervals,

each length s , in times t1 and t2 . Find the acceleration in terms of s , t1 , t2 .

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Phys 101 Midterm 1 ?? ??

Question 4 : Calculate the maximum possible acceleration amax .

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

1 kg

g = 10 m/sec

2

4 kg T

µµµ ==sk

sk µµ ,

F M m

g

FLIFT

θ

l T

100 N

10 kg

θ

g

µ= 3/4

2


Question 2 : Find the angle of projection at which the horizontal range is twice the maximum height reached by a

projectile.

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Question 3 : Find the tension on the cord.

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Question 4 : A mass M is whirled around in a vertical circle. The tension of the string at the bottom is twice the

tension at the top. Find Tb in terms of mg .

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Question 5 : A block slides down an inclined plane of inclination Ө with constant velocity . It is then projected up

the same plane with an initial speed v0 . How far up the incline will it move before coming to rest ? ₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 6 : A object with mass 2m and speed v explodes into two equal pieces in a gravity-free space. One piece

comes to rest. How much kinetic energy was added to the system by the explosion. ₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 H. Beker Fall '80

Question 2 : A train travels 32 km with a speed of 64km/hr, then it travels another 130 km with a speed 52 km/hr.

Calculate the average speed both in km/hr and m/sec.

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Question 3 : Two cars A and B are traveling in the same direction with

velocities 40 m/sec and 10/msec respectively. When car A is at a distance of 150

m behind car B, it starts to decelerate with deceleration “a” . Find the minimum

deceleration “a” to avoid collision of the two cars.

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Question 4 : A stone is thrown with an initial velocity of 50m/sec making an

angle of 37° with the horizontal. If the stone was thrown 160 m away from a

building which is 25 m high, find the range R on the roof the building.

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Question 1: A particle moving 1- D with constant acceleration passes two consecutive equal spaces, each of length

s , in times t 1 and t2 . Find the acceleration a in terms of s, t 1 and t 2 .

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µ=0.1

6 kg g T

2 kg

150 m

40m/sec

A B 10m/sec

50m/sec

37° 25m

160 m

R

3


Question 3 : Find the maximum possible horizontal acceleration that can be given to

the block shown in figure.

amax= amax(M,µ,g)=?

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Phys 101 Midterm 1 C. Saçlıoğlu , S. Nergiz Spring '92

Question 1 : Given ˆˆ ˆ3 - 2A i j k= +�

, find a unit vector n perpendicular to both A�

and the x-axis.

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Question 3 : A boy standing on a long, open train car throws a ball vertically

upwards with speed v with respect to the car . The car moves with constant

acceleration A all times: its speed at the instant the ball is thrown up is u with

respect to the ground. In which direction and how far must the boy move with

respect to the ground so that he can catch the ball?

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Question 3 : A rigid hoop of radius R rotates around a vertical axis with constant

angular velocity. A bead of mass m, which can move without friction along the hoop,

settles at an equilibrium angle θ as shown. Find;

a) the angular velocity , ω ,

the normal force N exerted by the hoop an the bead.

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Question 4 : a) Draw free body diagrams.

b) Find the minimum force F so that 1m will not slide down.

c) Find the acceleration, a.

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Phys 101 Midterm 1 S. Nergiz Fall '95

Question 4 : A small block is placed on a horizontal turntable of radius R, at a distance r

from its center. The turntable and the block are rotating with a constant angular speed ωωωω

about the center C of the turn table. At t = 0 the block mass has an initial position vector

iRrr ˆ)()0( +=�

relative to a coordinate system whose origin is located at the rim of

the turn table as shown in the figure. At an arbitrary time t find:

(i) the position vector of the block ?)( =tr�

, (ii) the velocity vector ?)( =tv�

and the

vector product ?=× vr��

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µ

M

g

θ

F= Mg

u

v

A

m

N

θN

ω

R

m1 m2

µs

F=?

g

µk

a=?

x

y

r

R

ωωωω

C 0

4


Question 4 : A small wedge of angle θ is rotated about a vertical axis with a constant

angular speed as shown in the figure. A small block of mass m is placed on the wedge

at a distance d from the top of the wedge. The coefficient of friction between the block

and the wedge is µs . Draw the free body diagram for the block and find the maximum

possible angular speed ω for which the block will stay at rest relative to the wedge.

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Question 4 : A small block of mass m1 is pushed against a larger block of mass m2

with a constant horizontal force F as shown in the figure. The coefficient of static

friction between the blocks is µs . There is no friction between m2 and the floor. (i)

Draw free body diagrams of each block and (ii) find the minimum value of the F to

hold m1 against m2 without sliding down.

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Question 4 : A train car is accelerating with a constant acceleration Âi−−−− relative to the

ground. In the train car a small block is placed on a wedge of angle θ . The coefficient

of kinetic friction between the block and the wedge is µk . Assuming that the block

slides upward on the wedge as shown in the figure, draw free body diagrams for the

block (i) in an inertial ground reference frame and (ii) in the normal car reference frame.

(iii) Find the acceleration vectors of the block in both reference frames.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3 : The velocity of a point moving along the x- axis is given by : x

Axv =)( . (everything is in MKS)

a) Unit of the constant A =? , b) a(x) = ? , c) x(t) = ? If x = 0 at t = 0 , d) v (t) = ? , e) a (t) = ?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 1 : A block of mass 5M rests on a frictionless floor. Another block of mass 2

M rests on it, connected to third block of mass 8M over a massless and frictionless

pulley. The coefficient of kinetic friction between 2 M and 5 M is 0, 25.

a) What are a5 , a2 , a8 ; the accelerations of 5M , 2 M , 8 M, respectively ?

b) What is the tension in the cord?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 2 : A hemispherical mound of mass M is being pulled by a force F over a

frictionless surface. A mass m rests on the mound at angle θ measured from the

vertical. The coefficient of static friction between m and M is µ. Find

a) Largest, b) The smallest values of F for which m does not move with respect to M.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

m

θ

d

µs

ω

g

F m1

m2

µs

µ=0

g

A

θ

m

µk

y

x

8M

2M

5M

µ=0.25 g

µ=0

m

g

θ µ=0

µ=µ

F

5


Question 3 : the potential energy function on a plane is given by 232),( yxyxU −= (MKS units)

a) Find F( x, y )�

,

b) Calculate the work done on a particle moving from (x,y) = ( 1,1 ) to (x,y) = ( 3,1 )

c) A particle of mass 2 kg starts at (x,y) = ( 1,1 ).with speed v = 8 m/s. What is its velocity at (x,y) = ( 3,1 ) ?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 F. Neyzi Spring '97

Question 1 : Given r�

rcosθ= i + rcosθ j and 2( )t tθ θ= = . (Every thing in MKS

units) Find v R and Tv , the radial and tangential components of the velocity at 2t ====

seconds.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 F. Neyzi Spring '97 Question 2 : Given 0 2s k .µ µµ µµ µµ µ= == == == = , what should M be so that the tension is 0.5 mg ?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3 : A The hollow cylindrical room in the figure is moving upward with

acceleration A while rotating at the same time around its axis. The coefficient of static

friction sµµµµ . The block remains pinned against the wall.

a) Draw a free body diagram,

b) What minimum rotational speed is necessary to prevent falling?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 4 : A person pushes a box of mass 20 kg up an incline as shown. F0 = 130 N

and a friction force of 40 N opposes the motion. Take g = 10 m/s2 .

(i) Draw a free body diagram,

(ii) Find the work done by each force

(iii) How fast will the box be going when it reaches the top if it starts from rest at the

bottom?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 Đ. Semiz Spring '98

Question 1 : A particle moves in a circle of constant radius b . If the speed of the particle varies with time t

according to the equation 2v At= , where A is a positive constant, for what value of t does the acceleration vector

make an angle of 45� with the vector?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

y

x θθθθ

m

M

T

0 2.µµµµ ====

T

sµµµµ

A

13m

5m

12m

F0 M

F0 : HORIZONTAL

6


Question 2 : A slab of mass 5m and length 2 meters rests on a frictionless

floor. A mass m rests on top of the slab as shown. The coefficient of kinetic

friction between the mass and the slab is µk = ½ .The mass m is acted on by a

horizontal force F = 3/2 mg such that it moves relative to the slab.

a) Draw the free body diagrams for both masses with respect to the

observer A at rest on the floor.

b) Find am and a5m accelerations of m and 5m respectively with respect to

the observer A at rest on the floor.

c) Find the acceleration of m as seen by observer B, at rest with respect to the slab.

d) Find the time t when the mass and the slab lose contact. (take g = 10 m/s2 ) ?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 1 : A ball of mass m is attached to a rigid vertical rod by means of two

massless strings as shown. The system is rotating about the axis of the rod, both

strings being taut (taut: tightly drawn, not loose) and the angles 1 2, θ θ and the

distance of the mass from the rod, R, are all known.

(a) Draw a free body diagram for m.

(b) If the tension in the upper string is 2

1 2

sin5

4 sin( )

mgT

θ

θ θ=

+ find the speed of the ball.

(c) Calculate the magnitude and the direction of the net force on the ball at the instant

shown.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 1 : A particle moves under the influence of a force ˆ ˆ2F kyi kxj= +�

from the

origin (0, 0) to point (a, a) . (a) Find the change in its kinetic energy if it moves along

the x-axis from (0, 0) to (a, 0) first and then to (a, a) .

(b) Find the change in its kinetic energy if moves along the line x = y.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 F. Neyzi, Ö. Oğuz, S. Turgay Spring '99

Question 1 : The position vector �r of a particle is given by

� ˆ ˆr A t i B sin u( t ) j= += += += + . u(t) is a function of time

given by 2 u( t ) C t==== . (A, B and C are positive constants). Find:

a) �v and

�a , the velocity and acceleration vectors respectively.

b) The angle between �v and

�a at t

Cππππ====

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 2 : Mass M1 rests on mass M2 inside an accelerating car as shown.

Blocks are to remain stationary relative to each other and relative to the car.

a) Draw a free body diagram for M1 and M2 with respect to a ground

observer.

b) What is the maximum acceleration A of the car?

c) What is the magnitude and direction of the force of friction on M1 at

this maximum acceleration?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

A

B

F=3/2mg

5meters

µ=1/2

µ=0

m

5m

R m

g=10m/sec2

T1

T2

1θ

2θ

y

x=y

(0,0) (a,0)

(a,a)

x

g=10m/sec2

M1

M2 µS2=1/4

µS1=1/2

A

M1=10 kg

7


Question 3 : A block of mass m rests on a frictionless incline, attached by a light

string as shown. The incline makes an angle θ with an elevator floor and the elevator

is accelerating upward with an acceleration A. Find T, the tension in the string.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 4 : A rotor is rotating with a constant angular velocity as shown. However

the speed is not enough to keep the person pinned up against the wall of the rotor and

he slides down only in the vertical direction. If the coefficient of friction between the

person and the wall is µ, find the magnitude of the net acceleration of the person with

respect to an outside observer. (You may express the net acceleration vector in terms

of unit vector r along radial direction and k vertical direction.)

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 Ö. Oğuz, S. Turgay Spring '00

Question 1 : The speed of a particle traveling in a circular path of radius R is given as 2v( t ) At==== , where A is a positive constant. At t = 2 seconds, find the acceleration

vector �a and the angle θθθθ , in terms of R and A.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 2 : Block M starts with an initial velocity 0v down an incline as shown in the figure.

After traveling a distance L, it hits an ideal massless spring with spring constant k.

The block comes to rest momentarily after it compresses the spring by D.

a) Find the coefficient of kinetic friction µ between M and the incline.

Find where (with respect to point A) the acceleration of the block changes direction.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3 : A force is given by 2 21 22

�ˆ ˆF( x, y ) k xy i k x y j= += += += + (k1 and k2 are positive

constants). Find the work done by this force when a particle moves from the origin to

point (3,1) : a) if it follows path 1 if it follows path 2

b) find the relationship between k1 and k2 if �F is conservative force

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 4: A block of mass M slides down on an incline making an angle θ with a

car floor as shown in the figure. The car moves with acceleration A to the right, with

respect to the ground. The coefficient of kinetic friction between M and the incline is

µ. Draw a free body diagram of M for:

a) an observer in the car b) an observer on the ground

Find the acceleration of M with respect to:

c) an observer in the car d ) an observer on the ground

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

T=? A

M

θ

µ

ω v = ωR

R

a�

= ?

θ = ?

v

R

g

k

M

v0

L

µ=?

point A

θ

y

B(0,3) 0 path 1

path 2 A(3,1)

g

M1

θ µ

A

8

Phys 101 Midterm 1 Sülün Turgay Spring '03 Question 3 : A box of mass m is pushed up a plane with inclination θ by a force

F that makes an angle of Φ with the surface of the plane: the kinetic friction

coefficient is µk. Draw the free body diagram of the box,

a) Find the acceleration of the box up the plane,

b) Find the work done by the friction force on the box if the height of the slope is

H.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 Sülün Turgay Spring '03 Question 4 : A slab off mass 3m and m are on the floor of a truck which moves

with acceleration. A = 0,5g as shown. The friction coefficients between the

surfaces are given by µ s = 0,6 µ’s = 0,4 ; µ’k = 0,3 .

a) Draw the free body diagram for each mass for an observer at rest on the

ground,

b) Does mass m slide on 3m? Why? Does mass 3m slide on the floor of the

truck? Why?

c) Find the acceleration of m as seen by observer at rest on the ground,

d) Find the value of the friction force between 3m and the floor.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 Sülün Turgay Spring '04 Question 1 : The system in figure is released from rest. The coefficients of kinetic and

static frictions are shown. The pulley is massless and frictionless.

a) Find the maximum value of m such that the system starts moving

b) assuming that block with mass m does not slide on block with mass 2M, find the

acceleration of the system for m = M.

c) Draw the inertial frame free body diagram of block with mass m and determine

whether the assumption in part b is correct.

d) determine the value of friction force between the blocks at this acceleration.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 Sülün Turgay Spring '04

Question 3 : Two objects A and B are thrown as shown in figure from the

origin. Find

a) The position vectors of the two objects as function of time and

b) The distance between the objects as a function of time ( Remember 37°

and 53° form 3-4-5 right triangle )

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 Sülün Turgay Spring '04

Question 4 : A particle moves in the x-y plane. Its position as a function of time is given by

2 2ˆ ˆr ( A Bt Ct )i ( D Et Gt ) j= + + + + += + + + + += + + + + += + + + + +�

where capital letters denote constants

a) Find the initial position vector of the particle.

b) Find the velocity vector as a function of time

c) Prove that the acceleration of this particle is constant.

d) A constant force F=Fj acts on this particle. Find work done by this force from time t=0 to time t=T

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 H. Beker Spring '05

Question 1 : A mass starts sliding down from the top of a frictionless hemispherical

mountain of radius R . Calculate the vertical and horizontal components of its velocity at

the instant it flies off the mountain.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

F

Φ H

m

θ

µk

3m

m

µ s=0,6 µk=0,5

µ's=0,4

µ'k=0,3

g A = g/2

µs=1/2

4M

µk=1/4

µ s=1

µk=1/2

F=3/2mg m

5M

y

53° 37°

5m/s

O

5m/s

x

A

B

R

9


Question 2 : Calculate v(t) if x

xv7

)( = and 4=x when 0=t

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3 : A frictionless inclined plane of inclination o30 , mass 2m and length ℓℓℓℓ is on a

frictionless table. Another mass m starts sliding from the top of the inclined plane.

Find υυυυ , the velocity of the inclined plane when m reaches the bottom.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 1 : A projectile is fired with a speed of 50m/s at an angle of 37° below

the horizontal from the top of a tower 80 m high. a) How much later will it hit

the ground ? b) What will be its horizontal range ?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 2 : One end of a 20 m long platform is slowly raised until a block

placed on it starts sliding at θ = 37° . Then the raising is stopped and it is

observed that the block reaches the bottom in 5 s. Determine µ s and µ k , the

static and kinetic coefficients of frictions.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3 : A block of mass 2 kg has a velocity of 150 î m/s ; another block of mass 3kg has a velocity of

– 40 î m/s. Calculate their final velocities after a one dimensional elastic collision

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 Đ. Semiz, S. Turgay Spring '06

Question 1 : The position of a particle with mass M is given by ˆ ˆ ˆr i j k= += += += +�

2 At + B sin(t) Ct where capital letters

denote constants. Find as a function of time; a) The velocity vector v�

, b) The acceleration vector a�

, c) The

force vector F�

, d) The kinetic energy K, e) The power P.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 2 : A small sphere of mass m is attached to end of a cord of length R and is set

into motion in a vertical circle about a fixed point O. For position shown: a) Draw free

body diagram of m with respect to a ground observer, b) Find tension in the string if m has

speed v at the given position, c) Find the acceleration vector, d) Find the rate of change of

speed.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3 : Masses m1 and m2 are on the floor of a truck which moves with

acceleration A = 0.5g as shown.

a) Draw free body diagrams for m1 and m2 .

b) Calculate the accelerations of m1 and m2 with respect to ground.

Find the direction and magnitude of the frictional forces acting on m1 and m2 .

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

m

l

2m

g

37° g = 10 m/s

2

sin37° = 0.6

37°

g = 10 m/s 2 sin37° = 0.6

R

O θ

=

m

g

g

µk = 0.6

µs = 0.8

A = 0.5 g

m1 m2

µk = 0.3

µs = 0.4

10


Question 4 : The net force acting on a particle with mass 2 kg depends on the position of the particle on the x-axis

according to the relation 5 2F x= − += − += − += − + (N)

The particle is initially at rest at the point x = 0 when the force begins to act.

a) Find the velocity of the particle as a function of position

b) After motion starts, find the position a such that v(a) = 0

Find the work done by that force as the particle moves from x = 0 to x = a

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 L. Kurnaz , S. Turgay Spring '07

Question 1 : A small block of mass m is at rest on the floor at the front of a railroad boxcar that has length

ℓ . The coefficient of kinetic friction between the floor of the car and the block is µ k. The car, originally at

rest, begins to move with acceleration a. The block slides back horizontally until it hits the back wall of

the car. (a) Draw the free body diagram of the block with respect to an inertial observer on the ground.

(b) As it reaches the back wall, what is its speed relative to the car? (c) relative to Earth?

(Given g, m, l, µ k, ,a)

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 2 : An amusement park ride consists of a large vertical cylinder that spins

about its axis fast enough that any person inside is held up against the wall when the

floor drops away. The coefficient of static friction between person and wall is µs, and

the radius of the cylinder is R. (Given µs, R, g)

(a) Draw the free body diagram of the person with respect to an inertial observer on

the ground.

(b) What is the maximum period of revolution necessary to keep the person from

falling.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3 : A ball of mass m is kicked of a cliff, H meters above the

ground, with an initial velocity ivv ˆ00 =

�. There is also a wind blowing

giving the ball an additional accelaration iAA windwindˆ=

�. (Given g, m, H,

v0, Awind)

(a) What is the final velocity v�

of the ball as it hits the ground?

(b) What is the change in the initial and final kinetic energies of the ball?

(c) What is the work done on the ball by gravity?

(d) What is the work done on the ball by the wind? ₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 4 : A horizontal force F is applied to the system shown so that

the masses move but m does not slide down on M. The coefficients of

static and kinetic friction between the two blocks is µs1 and µk1, between

block M and the ground is µs2 and µk2. (Given g, F, m, M, µs1, µk1, µs2, µk2)

(a) Draw the free body diagram of each of the masses with respect to an

inertial observer.

(b) Find the acceleration of the masses (given F).

(c) Find the minimum Fmin necessary so that block m does not slide. ₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

H

Awind v0

F M m

11


Question 1 : Workers pull a big rug from a bloc located 6m from the edge of the rug with a constant acceleration 8

m/sec 2. The coefficient of friction between the block and the rug is ½.

Draw a free body diagram for the block.

How far does the block move relative to the earth (take g=10m/sec2)

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 2 : A car rounds a banked curve. The radius of road is R, the banking angle is θ, the coefficient of static

friction is µs the coefficient of kinetic friction is µk Draw a free body diagram for the car and find the minimum

speed the car have without slipping up or down the road.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3 : ˆF kr= −= −= −= −�

, :r unit vector , where k is a constant, r is the distance from the origin to the point with

Cartesian coordinates (x,y)

a) Write the force in the Cartesian coordinates. b) Is it conservative? Explain. c) (extra credits) If yes, what is the

potential U(x,y) ? d) (extra credits) What is the potential U(r) ?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 4 : An object of mass 3,00 kg moving with an initial velocity of 5,00 i m/sec collides with and sticks to

an object of mass 2,00 kg with initial velocity of -3,00 j m/sec.

a)Find the velocity of the composite object.

b)Find the initial velocity of the 3,00 kg with respect to the center of mass.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 Y. SKARLATOS SPRING 08

Question 1: A child whirls a stone in a horizontal circle at a height h above the

ground by means of a string of length r. The string breaks, and the stone flies off

horizontally, striking ground at a distance d away. What was the centripetal

acceleration of the stone while in circular motion?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 2: Two identical birds sit on the middle of a stretched telephone wire

causing it to assume a V shape. Does this change the tension in the wire with

no birds on? If so, by an amount smaller than, equal to, or greater than the

weight of a single bird? Explain.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3 : Two blocks with masses M and m respectively are connected together

with a cord and placed on a horizontal frictionless table. The blocks

are pulled to the right with a force F acting on M. Calculate the tension

in the cord.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 4 : Find An object moves along the x-axis from x=1 to x=3 under the action of a force given by

F�

=2

A

x( ˆ î j+ ). Calculate the change in its potential energy.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

r

h

d

m M F T

12

Phys 121 Midterm 1 M. Mungan SPRING 08 Question 1: (Two questions) Your answers must be short but to the point!!!

a) Why do tires grip the road better on level ground then they do when going

uphill or downhill?

b) An evacuated box is at rest on a frictionless table as shown in the figure. You

punch a small hole in one face, so that air can enter. How will the box move?

What argument did you use to arrive at your answer?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 121 Midterm 1 M. Mungan SPRING 08

Question 2: In a home laundry dryer, a cylindrical tub containing the clothes is rotated

steadily about a horizontal axis. In order for the clothes to dry uniformly, they are made to

fall off the walls of the tube when they reach a certain angle θ, as shown in the figure.

Given that the radius of the tub is R, and the tub is rotating at an angular velocity ω, what

is θ?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3: Two ice sleds each of mass M are placed at a short distance

apart, one directly behind the other. A cat of mass m standing on one sled,

jump across to the other sled, lands on it, and immediately jumps back on to

the first. Both jumps are made at a speed of υ0 relative to the sled the cat is

standing on when the jump is made. Ignore friction between the ice and the sleds and find the final speeds of two

sleds.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 4: A man of mass m clings to a rope ladder suspended below a balloon of mass M, as

shown in the figure. The balloon is stationary with respect to the ground.

a) If the man begins to climb the ladder at a speed υ with respect to the ladder, in

what direction and with what speed with respect to the earth will the balloon

move?

b) What is the state of the motion after the man stops climbing?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 N. Inci , A. Kaya, F. Neyzi FALL 08 Question 1: A block of mass m1=20 kg accelerates with an acceleration a1=4

m/s2 on a block of mass m2=100 kg, which accelerates with a2=6 m/s

2 under a

force F on a frictionless table. The coefficient of kinetic friction between the

blocks is µ. Determine µ and F (g=10 m/s2).

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 N. Inci , A. Kaya, F. Neyzi FALL 08 Question 2: A particle of mass m is moving at the end of a string of length R. The

angular displacement of the particle is given by θ(t)=At2, where A is constant. In the

absence of gravity and friction, find the angle between velocity v�

and acceleration a�

as a function of time t. [Hint: Start with r Rr=�

ɵ and use the following definitions:

d r ˆdt

θθθθθθθθ=ɵɺ and

ˆdr

dt

θθθθθθθθ= − ɺ ]

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Vacuum

No friction

Air

m2

m1

µ F

a1

a2

θ

R

θ

13

Phys 101 Midterm 1 N. Inci , A. Kaya, F. Neyzi FALL 08

Question 3: A car of mass m moves on the side of a cone, which has static friction

coefficient µ. The axis of the cone is vertical, and gravity is directed downward.

The apex half-angle of the cone is θ, as shown in the figure. The path of the car

happens o be a circle of radius R in a horizontal plane. By drawing a free body

diagram, find the minimum value for the speed of the car such that it can drive

without slipping. ₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 N. Inci , A. Kaya, F. Neyzi FALL 08

Question 4: A force on a particle moving in the xy plane is given by 22 ˆ ˆF ( xyi x j )N= +

�, where x and y are in meters. The particle moves from

the origin to a final position having coordinates x=2 m and y=1 m as shown in

the figure. Calculate the work done by F�

on the particle as it moves along

(a) OAC and (b) OC ; (c) prove whether F�

is conservative or not.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 121 Midterm 1 A. Sevgen, FALL 08 Question 1: A car is accelerating to the right with a given acceleration A.

Find the minimum coefficient of static friction µs so that the block does not

slide.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 121 Midterm 1 A. Sevgen, FALL 08 Question 2: A wedge is accelerating to the right with a given acceleration A. There is a

block of m on the inclined plane(which is free to move on the wedge). Find the reaction

N on the mass m due to the wedge. Neglect friction.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 121 Midterm 1 A. Sevgen, FALL 08

Question 3: A mass m is on top of a curved shaped wedge of mass M and height h as shown

in the figure. Big mass M is free to slide on the ground. The small mass m is released from

rest, it slides down. After m loses contact with the wedge, its speed is measured to be v0 with

respect to the ground. Neglect friction. a) Find the velocity V of the wedge, after m lost

contact with it. b) Find the height h of the wedge. ₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 L. Akant, T. Turgut SPRING 09

Question 1: A particle’s position vector as a function of time t is given by 3 2( )r t t i t j tk= + +

� � � � (S.I)

Find a unit vector which is perpendicular to both velocity ( 1)v t =�

and acceleration ( 1)a t =�

.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 L. Akant, T. Turgut SPRING 09 Question 2: A projectile is fired from an initial height h

with an angle θ as shown in the figure. Find the initial

speed of the projectile if it has to hit a point located at

the height d from the ground and at a distance L from the

initial point.

R

v

θ

y

x A O

C(2,1)

g

m µs

A

m

A

g

θ

h

M

m

g

L

d h

θθθθ

υυυυ

g

14

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 L. Akant, T. Turgut SPRING 09 Question 3: A block of mass m is accelerated over a

rough surface by a light cord which passes over a light

pulley locate at the height h above and at a distance x

from the block. The magnitude of the tension is kept

fixed at the value T during the motion of he block. The

coefficient of kinetic friction between the block and the

surface is k

µ . Draw the free body diagram. Find the

acceleration of the block as a function of x,h, k

µ ,T,g

and m.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 L. Akant, T. Turgut SPRING 09 Question 4 A block of negligible size and of mass m rests inside a conical

surface as shown in the figure The conical surface rotates around its main axis

with angular velocity ω . The coefficient of static friction between the block

and the surface is s

µ . The block rests at the height above the apex of the

conical surface. The opening angle of the conical surface is θ. Find the

minimum and maximum ω so as to keep the block stationary at height h. Draw

the free body diagram in both cases.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 1: The resistive force acting on a skater is 2f kmv= − where k is a constant, m is the skater’s mass and v

is its speed.

a) If the initial speed is 0v , find the speed at any time t.

b) If the initial position is x=0, find the position x as a function of time.

c) Find the velocity as a function of position .

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 2: A block of mass 1m rests on top of a block of mass 2m which rests

on a frictionless table. The coefficient of static friction between the two blocks is

such that the blocks start to slip when the horizontal force applied to the lower

block is 0F . Suppose that a horizontal force is now applied only to the upper

block. What is its maximum value if the blocks are to slide without slipping

relative to each other? (Express your answer in terms of the given quantities only). For the case when the force is

applied to the lower mass, draw a free-body diagram for each mass separately.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3: Find the work done by the force 2 2( , 2 )F Ay Ax=

�� from (0,0) to (d,d)

along the path shown. Is the force conservative ? Can you find the potential U(x,y) ?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

x

h

g

m

h

T

ω

θ h

g

m2

m1

x

(d,0) (0,0)

y (d,d)

15


Question 4: A bullet of mass m and speed v passes through a

pendulum bob of mass M. The bullet emerges with a speed v/2.

What is the impulse on the bullet ?

The pendulum bob is suspended by a stiff rod of length l and

negligible mass. What is the minimum value of v such that the

pendulum bob will barely swing through a complete circle?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 H.Beker, B.Ünlü Midterm 1 Spring '10

Question 1: A projectile is thrown off a cliff as shown. Calculate the

time of flight, t and the range, R.

(g = 10 m/s2 , cos53° = 0.56, sin53° = 0.8)

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 H.Beker, B.Ünlü Midterm 1 Spring '10 Question 2 : M starts moving with respect to 5 M when F>2Mg. Calculate the coefficient

of static friction µs .

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 H.Beker, B.Ünlü Midterm 1 Spring '10 Question 3 : Two masses rotate about O on a horizontal

frictionless table with angular speed ω0 . Calculate the tensions T1

and T2 .

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 H.Beker, B.Ünlü Midterm 1 Spring '10

Question 4: A force acting on a particle is given by F = (3î + 4xĵ) N.

Calculate a) WAB the work done going from (0,0) to (2,2) b) WBC the

work done going from (2,2) to (0,2) c) WCA the work done going from

(0,2) to (0,0) e) WTOT the work done for the complete loop. f) Is the

force conservative?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

240 m

V0 = 50 m/s g

53

R=?

m M

O T2

2l

ω0

l

T1

y(m)

x(m)

A

A

(2,2) (0,2)

(0,0)

C B

F 5M

M

µs

µ = 0

m v

M

l

v/2

16


Question 1 : A ball of mass m swings into a wind exerting constant force F, on a rope

having length L and initially making an angle β with the vertical.

a. Draw a free-body diagram when m is at A.

b. Find the rate of change of speed, dv/dt, at A.

c. Find the work done by each force as m moves from A to B.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 121 Midterm 1 F.Neyzi Spring '10

Question 2 : A potential energy function for a two dimensional force is

U = x2y.

a) Find the force that acts at the point (x,y). Is it conservative? Why?

b) When the particle of initial energy E0 moves along a straigt line from (0,0) to

(a,b), where does it stop?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 121 Midterm 1 M. Sheftel Fall '10 Question 1 : An object of mass m1 on a frictionless horizontal table is connected to

an object of mass m2 through a massless pulley P1 and a massless fixed pulley P2 as

shown in figure.

a) If a1 and a2 are the accelerations of m1 and m2, respectively, what is the relation

between these accelerations?

b) Express the tension in the strings and,

c) The accelerations a1 and a2 in terms of the masses m1 , and m2, and g.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 121 Midterm 1 M. Sheftel Fall '10

Question 2 : A pendulum, comprising a string of length L and a small sphere, swings in

the vertical plane. The string hits a peg located a distance d below the point of

suspension.

(a) Show that if the sphere is released from a height below that of the peg, it will

return to this height after striking the peg.

(b) Show that if the pendulum is released from the horizontal position (θ = 900)

and is to swing in a complete circle centered on the peg, then the minimum value of d must be 3L/5.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 N. Đnci F. Neyzi Fall '10 Question 1 : An object of mass m experience a resistive force proportional to the square of its speed;

F = -mkv2 , where k is a constant. At t=0, x=0 and the speed is v0.

(a) Find the speed v for all times.

(b) Find the position x for all times.

(c) Find the object’s velocity as a function of position x.

(d) Find the object’s acceleration as a function of position x.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 N. Đnci F. Neyzi Fall '10

Question 2 : At t=0 and x2=d and both objects are at rest. At

time T, m1 losses contact with m2. Acceleration of m2 is a2 and

the coefficient of kinetic friction between the masses is µ.

a) Draw the free-body diagrams for all masses. b) Find the

magnitude and the direction of the acceleration of m1. c)

Find F. d) Find x1(t), t<T. (All answers must be in terms of

m1, m2, μ, g, T, d, and a2)

m

L β

F

B

A

g

(0,0) (a,0)

(a,b

)

x

x1

x2

m2

m1

a2

µ

F

g

m1

m2

P1

P2

Peg L

θ d

17

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 3 : The object m is attached to a vertical rod by two massless strings of

length l. The strings are attached to the rod at points a distance 2d apart. The

objects rotates in a horizontal circle at a constant speed. If the strings remain taut,

what is the minimum speed? Draw a free body diagram for m.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


Question 4 : A potential energy function for a system in which a two-dimensional force acts is of the form

U(x,y) = x2y – y,

a) Find the force that acts at the point (x,y).

b) Find the work done by this force as it moves from the point (0,0) to the point (a,b)

c) Is this force conservative? Why?

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

Phys 101 Midterm 1 Aktaş, Özcan, Beker Spring '11

1) The position of a particle is given by x = 4 + 3 sin(ππππ t) [ Everything is in SI units ] .

Determine a) its position when it changes direction for the first time and b) its velocity when it returns for

the first time to the position it had at t = 0 . ₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


2) A stone is thrown horizontally from the top of a cliff of height h toward the sea below.

It strikes the water at a distance d from the base of a cliff.

Calculate in terms of g , h , d : a) the time t at which the stone hits the water, b) the

initial speed of the stone, c) the speed of the stone just before it hits the water, d) the

direction of the stone's velocity just before it hits the water.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


3) A block of mass m is pushed up against a wall by a force P�

that makes an angle θ

with the horizontal. Calculate the minimum force minP for the block not to slide down.

State your answer in terms of the angle θ, the mass m, the gravitational acceleration g,

and the proper coefficient of friction between the mass and the wall. Check and explain

the validity of your result by evaluating its θ = 90o limit.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪


4) A truck is moving with constant acceleration ao up a hill that makes an

angle φ with the horizontal. Calculate ao , if a mass m hanging from the

truck's ceiling makes an angle θ with the perpendicular to the ceiling.

₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪

l

l

m 2d

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101 mid1 - Bogazici University Physics Department Phys 101 Midterm 1 H. Beker Fall '97 Question 3: the potential energy function on a plane is given by U(x, y) = −2x3 y2 (MKS units)