AP Physics C
Work and Energy
Free Response Problems with Calculus
1. A spring is constructed in which the spring force is proportional to the cube of the
displacement, where F = -kx3.This spring is placed on a horizontal frictionless surface.
One end of the spring is fixed, and the other end is fastened to a mass M. The mass is
moved so that the spring is stretched a distance A and is then released. Determine each
of the following in terms of k, A, and M.
a. The potential energy in the spring at the instant the mass is released.
b. The maximum speed of the mass.
c. The displacement of the mass at the point where the potential energy of the spring and
the kinetic energy of the mass are equal.
2. The above graph shows the potential energy U(x) of a particle as a function of its
position x.
a. Identify all points of equilibrium for this particle.
Suppose the particle has a constant total energy of 4.0 joules, as shown by the dashed
line on the graph.
b. Determine the kinetic energy of the particle at the following positions
I.
x = 2.0 m
II.
x = 4.0 m
c. Can the particle reach the position x = 0.5 m? Explain.
d. Can the particle reach the position x = 5.0 m? Explain.
e. On the grid below, carefully draw a graph of the conservative force
acting on the particle as a function of x, for 0 < x < 7 meters.
3. A particle of mass m moves in a conservative force field described by the potential
energy function U(r) = a(r/b + b/r), where a and b are positive constants and r is the
distance from the origin. The graph of U(r) has the following shape.
a. In terms of the constants a and b, determine the following.
I.
The position ro at which the potential energy is a minimum
II.
The minimum potential energy Uo
b. Sketch the net force on the particle as a function of r on the graph below,
considering a force directed away from the origin to be positive, and a
force directed toward the origin to be negative.
The particle is released from rest at r = ro/2
c. In terms of Uo and m, determine the speed of the particle when it is
at r = ro .
d. Write the equation or equations that could be used to determine where, if
ever, the particle will again come to rest. It is not necessary to solve for
this position.
e. Briefly and qualitatively describe the motion of the particle over a
long period of time.
4. A nonlinear spring is compressed horizontally. The relationship between the spring
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force and compression is given by the following formula: F = A 𝑥 2 , where x is the
distance from equilibrium that the spring is compressed and A is a constant. A group of
physics students used the spring on an experiment to investigate the elastic properties
of the spring. They recorded all collected data in the table below.
Force (N)
Compression
(m)
2.5
0.01
3.4
0.02
4.2
0.03
5.2
0.04
5.5
0.05
5.9
0.06
6.5
0.07
7.2
0.08
7.5
0.09
a. On the diagram below, make a graph that can be used to determine the constant A.
Be sure to label the x and y axis.
b. Find the constant A from the graph.
c. Develop and expression for the potential energy that can be stored in the spring.
The spring is used in a spring gun placed at the edge of a 0.75 m tall table. The
spring is compressed to a potential energy of 1.8 J. It then launches a 0.5kg dart.
d. Find the maximum distance from the table where a dart projected from the gun hits
the floor.
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5. An object is acted on by a force with a magnitude Fx = - 𝑥 2. The object moves along x
axis at a constant speed.
a. Calculate the work done by the force when the object moves from point x1
to point x2.
b. Calculate the work done by the external force when the object moves
from point x2 to point x1.
c. What is the difference between the answers in part (a) and (b)? What is
the net work done on the object?
6. A spring is placed on a horizontal table with one end fixed. The spring doesn’t obey
Hook’s Law and in order to stretch or compress it we need to apply the following force Fx
= (150 N/m)x – (650 N/m2)x2 + (12,500 N/m3)x3.
a. Calculate work required to stretch the spring from the equilibrium position
to x = 0.1 m.
b. Calculate work required to compress the spring from the equilibrium to x =
-0.1 m.
c. Compare your answers in part (a) and (b).
7. An object with a mass of 5 kg is initially at rest on a frictionless horizontal surface. A
horizontal force is applied to the object. The object changes its position as x = (0.4
m/s2)t2 + (0.04 m/s3)t3.
a. Calculate the velocity of the object at t = 5 s.
b. Calculate the kinetic energy of the object at t = 5 s.
c. Calculate force at t = 5 s.
d. Calculate work done by the force during first 5 s.
8. A spring exerts a restoring force Fx = -ax – bx2 when it is stretched or compressed.
Where, a = 50 N/m and b = 20 N/m. The mass of the spring is negligible.
a. Calculate potential energy of the spring as a function of position assuming
when x = 0, U(x) = 0.
b. Calculate work done by the spring when it stretches from x = 0 to x = 0.5
m.
c. An object with a mass of 0.5 kg is attached to the free end of the spring
and is placed on a frictionless horizontal table. The object is pulled to a
distance 0.5 m and released. Find the speed of the object when it is
located at x = 0.3 m from the equilibrium point.
9. A force exerted on an object is Fx = -αx2, where α = 15 N/m2.
a. How much work is done by the force on the object when it moves along a
straight line from x= 0 to x = 0.4 m?
b. How much work is done by the force on the object when it moves from
point x = 0.4 to point x = 0?
c. Is this force conservative? Explain.
d. If the force is conservative, find the potential energy, assuming when x =
0, U(x) = 0.
10. An object moving in the xy – plane is acted on by a conservative force. The potential
energy of the object is described by the formula: U(x,y) = a(x2 + y2) + bxy, where a and b
are positive constants.
a. Find the x component of the applied force.
b. Find the y component of the applied force.
c. Present the vector of force in terms of unit vectors.
Answer-Key:
1.
A.
𝑘𝐴4
4
𝑘
B. 𝐴2 √2𝑀
C. 𝐴⁄4
√2
2.
A. At x = 2, and at x = 5
B. I. 3J
II. 1J
C. No. The maximum energy is 4J, and at 0.5m, the energy is 6J. This cannot happen.
D. Yes. The maximum energy is 4J, and at 5m, the energy is around 3.8J. This is lower
than 4J, so it can reach 5m.
E.
3.
A. I. ro = b
II. 2a
B.
𝑈
𝑜
C. √2𝑚
D. 𝑈𝑟 = 𝑈𝑟 +
2
1
𝑚𝑣 2
2
E. The object will oscillate, but it will not be in simple harmonic motion.
4.
A.
F
10
9
8
7
6
5
4
3
2
1
1
𝑥2
.1
B. A ≈ 25 N𝑚2.5
𝐶.
2𝐴 1
𝑥2
3
D. 1.08m
.2
.3
.4
5.
A.
1
𝑥2
−
1
𝑥1
B.
1
𝑥1
−
1
𝑥2
C. (a) is negative, while (b) is positive. The net work done is zero.
6.
A. 0.85J
B. 1.28J
C. (a) is 0.43 less than (b)
7.
A. 7 m/s
B. 17.5J
C. 10N
D. 35J
8.
A. 25𝑥 2 +
20 3
𝑥
3
B. -7.1J
C. 4.32 m/s
9.
A. 0.32 J
B. -0.32 J
C. Yes. Work in both ways is equal in magnitude.
D.
10.
∝𝑥 3
3
A. -2ax – by
B. -2ay – bx
C. (−2𝑎𝑥 − 𝑏𝑦)𝑖̂ + (−2𝑎𝑦 − 𝑏𝑥)𝑗̂