**AP Physics – Work and Energy Example Problems**

1. A constant force of 10.0 N, 225° acts on an object as it moves 5.0 m, 120°. Determine the work done on the object by the force.

2. An
object moves on the *x*-axis beginning at *x* = 2 m and ending at *x*
= – 3 m. During this interval it is
subjected to a force given by *F*(*x*) = (4 N/m^{3})*x*^{3}.
Determine the work done on the object.

3. A
spring of length 50.0 cm is characterized by the constant *k* = 80.0
N/m. Determine the work required to
compress it from a length of 40.0 cm to 35.0 cm.

4. Starting from rest, a 5.0 kg sled is pulled across a level surface by an applied force of 27 N, 35°. This force acts as the sled is pulled a distance of 2.00 m. The coefficient of friction is 0.25. Determine the speed attained by the sled.

5. A
board 1.60 m long is propped up at one end by a table 0.60 m high. A 40__0__ gram block is started
sliding downward with speed 8.00 m/s at the top of the ramp and slides to the
bottom. The coefficient of friction is
0.30. (a) Determine the work done by each
force acting on the block. (b) Determine
the speed of the block at the end of the ramp.

6. A
cart of mass *m* is attached to springs as shown in the diagram below and
is free to move horizontally on the track where m is the
coefficient of friction. The springs
behave as a single spring with constant *k*. Suppose the cart is moved to a position *x*_{o}
relative to its equilibrium position (*x* = 0) and released from
rest. (a) Determine the speed of the
cart as it first passes equilibrium. (b)
Determine its position at which its speed first reaches zero after its
release. (c) Determine how many times
the car will pass by *x* = 0 before coming to a stop.

7. The mass of the space shuttle is 79000 kg and it orbits at a uniform altitude of 350 km. Ignore air resistance and the rotation of the earth. (a) Determine the net work required to put the shuttle into this orbit. (b) In order to return to earth, the shuttle fires retrorockets that reduce its orbital speed by 91 m/s. Determine the speed of the space shuttle when it lands.

8. The
actual landing speed of the space shuttle is 1__0__0 m/s. Use relevant information from the previous
problem. (a) Determine the work done by
the earth’s atmosphere during the shuttle’s return to the surface. (b) Estimate
the amount of work done by the rocket engines to put the shuttle into
orbit. (c) Conceptual question: What difference does the earth’s rotation
make? What would you do differently in
these calculations if you wanted to account for it?

9. The
gravitational field inside the earth may be modeled by *g* = a *r*^{2}
+ b
*r*, where *r* = distance from center, a = – 2.851 ´ 10^{-13}
m^{-1}s^{-2},
and b
= 3.354 ´
10^{-6}
s^{-2}. (This model allows for the increasing density
towards the earth’s core using r(*r*) = (-0.00136 kg/m^{4}) *r*
+ 12000 kg/m^{3}). (a) Use the
function *g*(*r*) to determine the potential energy function for an
object located inside the earth. Use the
earth’s center as a reference point. (b)
Supposing an object could fall through a tunnel from the surface to the center
of the earth what would be its final speed?
(c) What would be the escape speed for an object to be shot out of this
tunnel and leave Earth’s gravity?

10. A particle
of mass 2.00 kg is free to move along the *x*-axis. The particle’s potential energy is given by *U*(*x*)
= *x*^{3} − 3*x* + 3, where *x* is position in meters and *U*
is energy in joules. The particle begins
at rest at the origin. (a) Determine the
particle’s maximum speed. (b) Determine
the particle’s greatest departure from the origin. (c) Determine the particle’s maximum rate of
acceleration (in either direction). (d)
Describe in words the overall motion of the object. (e) What initial speed would be required to
prevent the particle from returning to the origin? (f) If the particle is given speed 3.00 m/s
at the origin what will be its speed at *x*
= −3 m?

Answers:

1. −13 J

2. 65 J

3. 0.500 J

4. 3.3 m/s

5. a.
friction: −1.7 J; gravity: 2.35 J;
normal: 0 J

b. 8.2 m/s

6. a.
*v* = (*kx*_{o}^{2}/*m*
− 2μ*gx*_{o})^{0.5}

b. *x* = 2μ*mg*/*k* − *x*_{o}

c. integer part of *n* = *k*|*x*_{o}|/2μ*mg*

7. a.
2.6 × 10^{12} J

b. 8020 m/s

8. a.
−2.5 × 10^{12} J

b. 5.1 × 10^{12} J

c.

9. a.
*U*(*r*)
= *m*(a *r*^{3}/3 + b *r*^{2}/2)

b. 9.33 km/s

c. 14.6 km/s

10. a. 1.41 m/s

b. 1.73 m

c. 3.00 m/s^{2}

d.

e. 1.41 m/s

f. 5.20 m/s