AP Physics Assignment – Equilibrium and Oscillation

Reading            Physics for Scientists and Engineers   pp. 337 – 346, 418 – 435

Objectives/HW

 The student will be able to: HW: 1 State and apply the conditions for a particle or rigid body to be in equilibrium and solve related problems. 1 – 12 2 State and apply the condition for stable equilibrium and contrast with unstable equilibrium and solve related problems. 13, 14 3 Solve problems involving Simple Harmonic Motion including those concerning:  conditions for occurrence, relation between period and the force constant k, relation between period and angular frequency, analyses of position, velocity, and acceleration using sine and cosine. 15 – 24 4 Solve problems involving simple pendulums. 25 – 27 5 Solve problems involving physical pendulums. 28 – 30

Homework Problems

1.      A tightrope walker of mass 80.0 kg stands at rest in the center of the cable.  The cable forms an angle of 170.0° at the point where he stands, as shown in the diagram below.  What is the tension in the cable?  Ignore the mass of the cable.

2.      A picture frame of mass 200.0 g is hung by a wire attached to its back.  The wire is 35.0 cm long and is attached at either side of the frame which is 30.0 cm wide.  Find the tension in the wire when the frame is hung from a nail on the wall.  Ignore friction.

3.      Find the tension in each wire supporting the 100.0 N weight shown in the diagram below.

4.      Find the tension in each wire supporting the 100.0 N weight shown in the diagram below.

5.      For each pulley system shown below, what amount of force must be applied to lift the 100.0 N weight at a constant velocity?  Disregard the mass of the string and of the pulleys.

(a)                     (b)                           (c)

6.      A Mini Cooper has a wheelbase of 97 inches (distance from front axles to rear axles).  When parked on a level surface, the normal force on each front tire is 820 pounds and the normal force on each rear tire is 490 pounds.  (a) What is the weight of the vehicle?  (b) How far back from the front axle is the center of mass?

7.      The center of mass is located at a position above the ground 0.56 times the height of a typical male human.  How much force does it take to do a pushup?  Consider a man that weighs 175 lbs and is 6.0 feet tall.  When doing a pushup, he puts all of his weight on his toes and his hands, which are placed even with his shoulders – 5.0 feet from his toes.  (a) What is the total force he must exert with his hands?  (b) What is the total amount of force on his toes?  (c) Repeat for a female with the same numbers except the center of mass is 0.54 times height.

8.      Two men move a heavy dresser up some stairs that climb at an angle θ above horizontal.  The dresser has weight W, height h, and length L.  The men grasp the dresser along its bottom at each end.  Determine an expression for the required force of each person to move the object at constant velocity up the stairs.  Assume the center of mass of the dresser is at its geometric center.

9.      A kid pulls a wagon that weighs 200.0 N across a level surface at constant speed by applying a force of 40.0 N, 50.0°.  The rope is attached at a point 0.400 m above the ground and directly above the front axle.  The rear axle is 0.900 m from the front axle. The center of mass of the wagon is midway between the axles.  (a) Find the total friction on the wagon.  (b) Find the vertical force at the rear axle.  (c) Find the vertical force at the front axle.

10.  A uniform beam of mass 100.0 g is attached to the wall with a hinge and supported at the other end by a cable as shown in the diagram below.  A mass of 1.50 kg is hung from the end of the beam.  (a) Find the tension in the cable.  (b) Find the force on the wall at the hinge.

11.  A ladder of mass 10.0 kg and length 3.00 m is propped against a vertical wall.  The bottom of the ladder is 1.00 m from the base of the wall.  Friction between the top of the ladder and the wall is negligible.  (a) Find the amount of force the ladder exerts on the wall.  (b) Find the amount of friction at the bottom end of the ladder.  (c) Find the amount of normal force on the bottom end of the ladder.

12.  A mass of 15000 kg is supported by the structure shown below.  (a) Determine the forces that act on the structure at points A and B.  (b) Determine the load in each beam.

13.  For each of the following functions describing the net force on a particle, determine any and all positions of equilibrium and characterize each as stable or unstable.  (a) F = 4x − 2x3, (b) F = x2 + x − 6,  (c) F = 5x − 9.

14.  Suppose the particle in the previous problem is at rest at one of the equilibrium points.  It is then nudged slightly to the right and released.  (a) Describe in words the resulting motion if it is a point of stable equilibrium.  (b) Describe in words the resulting motion if it is a point of unstable equilibrium.

15.  A mass of 200.0 grams is hung from a vertical spring with constant k = 30.0 N/m.  The mass is pulled downward 10.0 cm from its equilibrium position and released.  (a) Find the period of the resulting oscillation.  (b) Find the maximum speed of the mass.  (c) Determine the velocity and acceleration of the mass at a point in time precisely 3.00 seconds after its release.  (d) Determine the velocity and acceleration of the mass when it is located 15.0 cm above the point of release.

16.  A mass on the end of a spring is observed to oscillate 10.0 complete cycles in 14.0 seconds.  The amplitude of its oscillation is 5.00 cm.  (a) Find the maximum speed of the mass.  (b) Find the maximum rate of acceleration.  (c) What is the ratio of k/m?

17.  A mass of 500.0 g on the end of a spring is observed to move back and forth with a frequency of 2.00 Hz.  During this motion its maximum speed is 4.00 m/s.  (a) Determine the spring constant k.  (b) Find the amplitude of its oscillation.

18.  A certain car of mass 1500 kg bounces up and down on its springs with a frequency of 1.50 Hz.  (a) Find the effective spring constant of the four springs combined.  (b) Determine the spring constant of one of the four springs found at each wheel.  (Hint:  assume the springs are identical and each spring is compressed an equal amount.)

19.  A car’s shock absorbers are designed to dissipate energy so that it will not continue to bounce up and down on its springs after hitting a bump.  (a) Suppose the car in the previous problem encounters a bump that causes the springs to be compressed 10.0 cm – how much energy must the shock absorbers dissipate?  (b) If this energy is not absorbed (i.e. the shock absorbers are removed) what will be the maximum speed of the bouncing car?

20.  A certain object of mass 0.50 kg moves along the x-axis.  Its potential energy is given by U = 3x2 + 8x.  Starting at rest, the object is released at the origin.  (a) Find the maximum speed attained by the object.  (b) Find the position about which it oscillates.  (c) Find the frequency of the oscillation.

21.  A particle of mass 2.0 kg moves along the x-axis subject to a net force of F = 2 + 5xx2, where F is in newtons and x is in meters.  (a) Find a position of stable equilibrium about which the particle can oscillate.  (b) Determine the period assuming the amplitude of the oscillation is very small.  (Hint: a “small piece” of this function can be viewed as “approximately linear” with a particular slope – use this idea to find a value of k.)

22.  A lab cart of mass 725 grams rolls on a track at speed 4.00 m/s.  It runs into a horizontal spring with constant k = 50.0 N/m mounted at the end of the track and bounces off of it.  For how much time is the cart in contact with the spring?

23.  The axle of a solid wheel of mass M and radius R is attached to a horizontal spring of constant k as shown below.  Derive and simplify an expression for the period of the wheel’s oscillation, assuming it rolls back and forth without slipping.

24.  A small marble is placed in the bottom of a large bowl.  If the marble is given a small push it will oscillate back and forth about the bottom of the bowl, moving along a small arc with radius 0.50 m.  Assuming the marble rolls without slipping what is the period of this oscillation?  Hints:  use the moment of inertia for a solid sphere and the small angle approximation q  = sin(q ) – this is true if the angle is small and measured in radians.

25.  A pendulum is formed by hanging a small mass of 50.0 grams on the end of a string that is 2.50 m long.  The mass is pulled back so that the string forms a 5.0 º angle with the vertical and then it is released.  (a) Find the period of this pendulum.  (b) Find the angular frequency.  (c) Determine the maximum angular velocity that occurs during its swing.  (d) Determine the maximum linear acceleration that occurs during its swing.

26.  A spelunker rappels a long rope down a vertical shaft in a cave.  He stops and rests, hanging on the rope.  In order to estimate how far down the rope he has descended, he pushes off a nearby rock and swings like a pendulum.  (a) If he swings with a period of 10.0 s, how far down the rope is he?  (b) Determine a formula that could be used to easily approximate the length in such a circumstance.  (One that you could use “in your head”.)

27.  One way to determine the value of g is to measure the period and length of a pendulum.  (a) Suppose a simple pendulum of length 50.0 cm has a period measured to be 1.44 s.  What is g based on these values?  (b) A better method is to vary the length and period and produce a graph of period vs. the square root of the length.  Suppose the slope of this graph is 2.010 s m1/2.  What is g based on this graph?

28.  A meter stick is pivoted about one end so that it can swing like a pendulum.  What is the period?

29.  A kid hangs her hula-hoop up on a nail in the wall.  It hangs there wobbling on the nail as she walks away.  If the hoop has a diameter of 90.0 cm, what is the angular frequency of its wobbling?

30.  A grandfather clock has a pendulum formed by a rod of length 80.0 cm and mass 100.0 grams with a solid disk of mass 200.0 g and radius 6.00 cm.  The plane of the disk is parallel to the plane of the pendulum’s oscillation and its center is attached to the very end of the rod.  (a) Determine the period of the pendulum.  (b) If the clock runs slow by 1.00 minute per a 24 hour period, how much further up the rod should the disk be attached?

1.      4.50 kN

2.      1.90 N

3.      T1 = 119 N, T2 = 156 N, T3 = 100 N

4.      T1 = 81.5 N, T2 = 92.2 N, T3 = 100 N

5.      a. 50.0 N
b. 25.0 N
c. 25.0 N

6.      a. 2620 lbs
b. 36 in

7.      a. 118 lbs
b. 57 lbs
c. 113 lbs, 62 lbs

8.      guy in back
guy in front

9.      a. 25.7 N, backward
b. 88.6 N, up
c. 80.8 N, up

10.  a. 35.9 N
b. (32.6 i + 0.49 j) N (or 32.6 N, 0.9º)

11.  a. 17.3 N
b. 17.3 N
c. 98 N

12.  a. FA = 201 kN, down, FB = 348 kN, up
b. AC = 284 kN tension
AB = 201 kN compression
BC = 402 kN compression

13.  a. x = 0, unstable; x = ±1.41, both stable
b. x = −3, stable; x = 2, unstable
c. x = −1.8, unstable

14.  a.
b.

15.  a. 0.513 s
b. 1.22 m/s
c. 1.00 m/s, down; 8.64 m/s2, up
d. 1.06 m/s, up or down; 7.50 m/s2, down

16.  a. 0.224 m/s
b. 1.00 m/s2
c. 20.1 s−2

17.  a. 79.0 N/m
b. 0.318 m

18.  a. 130 kN/m
b. 33 kN/m

19.  a. 670 J
b. 0.94 m/s

20.  a. 4.6 m/s
b. x = −1.3 m
c. 0.551 Hz

21.  a. x = 5.4 m
b. 3.7 s

22.  0.378 s

23.

24.  1.7 s

25.  a. 3.17 s
c. 0.173 rad/s or 9.90 deg/s
d. 0.855 m/s2

26.  a. 24.8 m
b.

27.  a. 9.52 m/s2
b. 9.77 m/s2

28.  1.64 s