Chapter I - College of Engineeringjfrhoads/CalDynamics/H1_cal_summer_2016.pdfChapter I: Particle Kinematics Homework Homework 1.A Given: Particle P moves within the x-yplane on a path

Chapter I: Particle Kinematics Homework

Chapter I

Particle Kinematics Homework

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Homework 1.A

Given: Particle P moves within the x-y plane on a path described by y (x) = 2√

4x2 + 12 whereboth x and y are given in feet.

Find: Determine the acceleration of P. Write your answer as a vector in terms of its x-y components.

Particle Kinematics Homework Problems ME 274

x

y path of P

P

Problem I/1

Particle P moves within the x-y plane on a path described by y x( ) = 2 4x2 +12 , where both x and y are given in feet. Determine the acceleration of P. Write your answer as a vector in terms of its x-y components. Use the following parameters in your analysis: x = 6 ft , !x = !2 ft / sec and

!!x = !3 ft / sec2 .

Use the following parameters in your analysis: x = 2 ft, x = -10 ft/s and x = 0.



Homework 1.B

Given: An automobile A is traveling on a circular path centered at O and having a radius of R.The automobile has a speed of v and is changing this speed at a rate of v .

Find: For this problem:

(a) Determine the acceleration of A. Write this as a vector in terms of its x-y components.

(b) Make a sketch of the acceleration vector for A.

(c) Determine the magnitude of the acceleration of A in terms of the number of “g’s” experiencedby a passenger in the automobile.


Problem I/2 An automobile A is traveling on a circular path centered at O and having a radius of R. The automobile has a speed of v , and is changing this speed at a rate of !v .

a) Determine the acceleration of A. Write this as a vector in terms of its x-y components.

b) Make a sketch of the acceleration vector for A. c) Determine the magnitude of the acceleration of A in terms of the number of “g’s”

experienced by a passenger in the automobile. Use the following parameters in your analysis: R = 150 m , ! = 53.13° , v = 20m / s and

!v = !2m / s2 .

x

y

A

v

O

R

!

Use the following parameters in your analysis: R = 75 m, θ = 135◦, v = 10 m/s and v = -6 m/s2.



Homework 1.C

Given: A skier is moving down a slope. Along the straight slope from A to B, her speed is changingat a constant rate of vAB. From B to C, the slope is curved with a radius of curvature of ρ. Atpoint C, her speed is changing at a rate of vC . The skier’s speed at points A, B and C are vA, vBand vC , respectively.


(a) Sketch the tangent and normal unit vectors at points A, B and C.

(b) Calculate the acceleration vectors at points A, B and C.

(c) Sketch the acceleration vectors at points A, B and C.

A

B

C

Use the following parameters in your analysis: vAB = 5 m/s2, vC = -1.5 m/s2, ρ = 175 m, vA =20 m/s, vB = 30 m/s and vC = 40 m/s.



Homework 1.D

Given: A particle P moves within a plane with a path given in terms of the polar coordinates:R = 2θ2 where R and θ are as defined below in the figure, and with R and θ given in meters andin radians, respectively.


(a) Determine the velocity and acceleration vectors of P in terms of their polar coordinates.

(b) Make a sketch of the velocity and acceleration vectors for P.


Problem I/4 A particle P moves within a plane with a path given in terms of the polar coordinates of:

R = 2!2 where R and ! are as defined below in the figure and with R and ! given in meters and in radians, respectively.

a) Determine the velocity and acceleration vectors of P in terms of their polar coordinates.

b) Make a sketch of the velocity and acceleration vectors for P.

c) Determine the rate of change of speed of P.

d) Is speed of P increasing or decreasing at this instant? Explain.

Use the following parameters in your analysis: ! = " radians ,

!! = " 1.5 rad / sec and

!!! = 2 rad / sec2 .

R

P

O

!

eR

e!

Use the following parameters in your analysis: θ = π/2 rad, θ = -3 rad/s and θ = -2 rad/s2.



Homework 1.E

Given: A particle P moves in three-dimensional space on a surface given by z = x2y, wherex = t2/2 and y = 3 cos 2πt.

Find: For time t = 1 s, determine the velocity and acceleration vectors of P in terms of theirCartesian coordinates.



Homework 1.F

Given: A particle P moves within a plane with a path given in terms of the polar coordinates of:R = 2θ2, where R and θ are as defined below in the figure, and with R and θ given in meters andin radians, respectively.


(a) Determine the velocity and acceleration vectors of P in terms of their polar coordinates.

(b) Make a sketch of the velocity and acceleration vectors for P.

(c) Determine the rate of change of speed of P.

(d) Is the speed of P increasing or decreasing at this instant? Explain.


Problem I/4 A particle P moves within a plane with a path given in terms of the polar coordinates of:

R = 2!2 where R and ! are as defined below in the figure and with R and ! given in meters and in radians, respectively.

a) Determine the velocity and acceleration vectors of P in terms of their polar coordinates.

b) Make a sketch of the velocity and acceleration vectors for P.

c) Determine the rate of change of speed of P.

d) Is speed of P increasing or decreasing at this instant? Explain.

Use the following parameters in your analysis: ! = " radians ,

!! = " 1.5 rad / sec and

!!! = 2 rad / sec2 .

R

P

O

!

eR

e!

Use the following parameters in your analysis: θ = π/2 rad, θ = -3 rad/s and θ = -2 rad/s2.



Homework 1.G

Given: Automobile A is traveling down a roadway with a speed of vA when it encounters ahailstorm. Hailstone P is known to be falling with a speed of vP at an angle of θ forward of theautomobile.


(a) Write down, as a vector, the velocity of P as seen by a passenger in the automobile.

(b) At what angle with the vertical is this observed velocity of P?


P

A

vA

vP !

Problem I/5 Automobile A is traveling down a roadway with a speed of vA when it encounters a hailstorm. Hailstone P is known to be falling with a speed of vP at an angle of ! forward of the automobile.

a) Write down, as a vector, the velocity of P as seen by a passenger in the automobile.

b) At what angle with the vertical is this observed velocity of P?

Use the following parameters in your analysis: vA = 20 m / sec , vP = 6 m / sec and ! = 53.13° .

Use the following parameters in your analysis: vA = 40 m/s, vP = 10 m/s and θ = 20◦.



Homework 1.H

Given: Two cars are travelling as shown. Car A is traveling along a circular path of radius, r,with a constant speed, vA. Car B is traveling with a speed, vB, in the direction of the positivex-axis. The speed of B is changing at a rate of vB.

Find: At this instant:

(a) Determine the velocity of A as seen by car B.

(b) Determine the acceleration of A as seen by car B.


Problem I/14 Two cars are travelling as shown. Car A is traveling along a circular path of radius, r, with a constant speed, Av . Car B is traveling with a speed, Bv , in the positive x axis. The speed of B is changing at a rate of Bv! . At this instant, find:

a) The velocity of A as seen by car B. b) The acceleration of A as seen by car B.

Use the following parameters in your analysis: r = 100 m, Av = 15 m/s, Bv = 20 m/s,

Bv! = -2 m/s2.

x

y

r

A

B

Use the following parameters in your analysis: r = 150 m, vA = 20 m/s, vB = 30 m/s and vB = 4m/s2.



Homework 1.I

Given: A worker is lifting a large crate, P, using a pulley shown in the diagram. He is holding therope at point A and is walking to the right at a constant speed vA.

Find: What is the speed of the crate when the worker has walked a distance x?


Problem I/15 A worker is lifting a large crate, P, using a pulley shown in the diagram. He is holding the rope at point A and is walking to the right at a constant speed Av . What is the speed of the crate when the worker has walked a distance x? Use the following parameters in your analysis: Av = 1.5 m/s, x = 2 m, h = 5 m.

A A’ P

P’

x

h

O

Use the following parameters in your analysis: vA = 2 m/s, x = 3 m and h = 8 m.



Homework 1.J

Given: Blocks A and B are connected by the pulley-system shown below. Block A moves downwardwith a constant speed of vA on a vertical guide. Assume the radii of the pulleys to be small.


(a) Determine the speed of block B when sA = 0 m.

(b) Determine the speed of block B when sA = 4 m.


Problem I/6 Blocks A and B are connected by the pulley-system shown below. Block A moves downward with a constant speed of vA on a vertical guide. Assume the radii of the pulleys to be small.

a) Determine the speed of block B when sA = 0 .

b) Determine the speed of block B when sA = 4 m . Use vA = 18 m / sec in your analysis.

B A

sA sB

1.5 m

O

vA

Use the following parameters in your analysis: vA = 25 m/s.


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Chapter I - College of Engineeringjfrhoads/CalDynamics/H1_cal_summer_2016.pdfChapter I: Particle Kinematics Homework Homework 1.A Given: Particle P moves within the x-yplane on a path