Homework 4

PHY211 - 2015 Summer Session II

Due date: 08/5/2015

Your homework solutions should be easily legible. You may collaborate with fellowstudents, but write up solutions independently.

1 Problems from Knight.

1. Page 213, problem 48

2. Page 286, read example 11.5

3. Page 305, problems 42 and 48 (only part a.)

2 Free fall

An object of mass m = 2.1 kg falls from an height of 20 m to the ground. Calculatethe work done by the force of gravity between the starting point and the end pointof the trajectory.

3 Spring

A massless spring of elastic constant k = 0.2 N/m has a length of 10 cm if no forceis acting on it. Now a toy car is pushed against the spring in such a way that thelength of the spring is 7 cm. Suppose you hold the car so that it doesn’t move. Thereis no friction in this problem.

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a. What is the potential energy stored in the spring? What is the Force that youhave to exert on the car in order to keep it in its position? Choose a positivedirection for the x axis and indicate direction and magnitude of the force.

Now suppose you instantaneously remove your hand.

a. At this instant of time, what is the direction of the acceleration of the car?What is the magnitude of the acceleration?

b. The car starts, say, in position x = 0m, and accelerates, say, towards right.Will the force on the car be constant or will it change with position while thecar is pushed to the right? At what value of the x coordinate will the force onthe car be zero?

c. What is the value of the acceleration of the car when the length of the springreaches 10 cm? What is the speed of the car at this point?

d. Suppose that the car can freely disconnect from the spring. At what positionwill it loose contact with the spring? Why? [Hint: consider the velocity of thecar and of the end of the spring during the motion.]

4 Work-Kinetic energy theorem

Apply the Work-Kinetic energy theorem to the problem of parabolic motion: consideran object launched from the level of the ground with an initial velocity |v0| = 3 m/sat an angle of 45 degrees above horizontal. Call vf the velocity of the object whenit hits the ground. Set the zero of potential energy at the ground level.

a. What is the work done by gravity from the starting point (0,0) to the top ofthe trajectory? (be careful: the top of the trajectory is the point where theheight of the object is maximum.)

b. What is the work done by gravity from the top of the trajectory down to theground?

c. What is the total work done by the gravitational force on the object?

d. Now that you know the value of the total work, use the work energy theoremto find the magnitude of the final velocity vf .

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5 Conservation of energy 1

A stone of mass m = 0.2 kg is connected to a string of length l = 1 m, and it isspinning in a vertical circle. Don’t ignore the presence of the gravitational force.

a. What is the symbolic expression of the total energy of the stone at any pointof the trajectory? Use m, g, l, θ, vT to express your result.

b. Calculate the minimum value of the tangential velocity at the top needed tostay in a circular trajectory.

c. Making use of the result of the previous question, use conservation of energyto find the tangential speed of the stone at the bottom.

d. What is the work done by the force of gravity between point A and B? (seepicture). What is the work done by the force of gravity between point B andC? (Be careful with the ± sign) [Hint: use the work-energy theorem to relatework with EK , and then use conservation of energy to calculate ∆EK .]

6 Pendulum

A pendulum of length l = 2 m and mass m = 0.2 kg is swinging from the ceiling.The amplitude of the oscillation is θmax = 10 degrees.

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1. Compute the total force acting on the bob at the lowest point P of the trajec-tory. Give magnitude and direction.

2. Compute the tangential acceleration of the object at the lowest point P of thetrajectory. [Hint: use the second law.] Can we say that instantaneously this isa uniform circular motion?

3. If the bob was just hanging from the ceiling without swinging, would the tensionin the rope be less than, greater than, or equal to the tension you found inquestion 2? Why?

7 Conservation of Energy 2: Pendulum

A mass m = 0.1 kg is hanging from the ceiling through a massless string of lengthl = 1 m. The mass is raised up to the point where the string makes an angle θ = 20degrees with the vertical direction, and released at the time t = t1.

a. Draw the free body diagram of the mass at time t = t1. Draw the componentof mg perpendicular to the string. Draw the component of mg in the directionof the string. Compute the values of the components using the data of theproblem.

b. Compute the tangential speed of the mass at the bottom of the trajectory usingconservation of energy. Place the zero of potential energy at the level of thelowest point.

8 Equilibrium + Energy conservation

A system of two identical masses m = 0.3 kg is suspended from the ceiling as shownbelow:

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The lengths of the ropes are: l1 = 0.7 m, l2 = 0.6 m. The distance d is 2 m. Findthe tension in the ropes (magnitude and direction). [Hint: indicate all the relevantangles that you are going to use to decompose the forces into components.]The rope l2 is cut. What is the tension in the ropes? Use conservation of energy tofind the speed with which the masses hit the walls.

9 Conservation of energy and Heat

In its general form, the conservation of energy involves mechanical energy (kinetic+ potential), and thermal energy. When we deal with sliding objects, and there iskinetic friction, the force of friction itself does Work on the moving object.

a. Take an object of mass m = 10 kg sliding on a surface where the coefficientof kinetic friction is µk = 0.4. Suppose that initially the block is moving atconstant speed v1 = 2 m/s (an external force is balancing the friction). Then,at position x1, the external force disappears. Compute the acceleration of theobject, and the length travelled before it comes to a complete stop at positionx2. Then, calculate the work done by the force of friction between x1 and x2.This work is the amount of energy that has been dissipated as heat.

b. Compute the kinetic energy at x1 and at x2. Verify that the Work-energytheorem is valid.

This exercise is an example of conservation of total energy (mechanical + thermal),which can be stated as:

EK +Wfriction = const.

WhereWfriction is the amount of heat dissipated through friction. SometimesWfriction

is also denoted by Q.

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10 Extra Problem: Potential Energy

(You don’t have to solve this problem, but if you do, you can get 5 extra points.)Find a symbolic expression for the potential energy of a spring Uel(x), using integrals.The force of the spring is

F = −kx (1)

[Hint: use the integral definition of work:

W =

∫ x2

x1

F · dx . (2)

Notice that in this problem force and displacement are parallel, so first write downF · dx as a scalar product, and then rewrite the integral as

W =

∫ x2

x1

F (x) dx . (3)

Where F (x) is the component of the force (1) along x. Now, substitute F (x) from (1)and calculate the integral.]

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