Work and Energy
1. CONCEPTUAL PROBLEMS
1.1. True or false:
(a) Only the net force acting on an object can do work.
(b) No work is done on a particle that remains at rest.
(c) A force that is always perpendicular to the velocity of a particle
never does work on the particle.
1.2. You are to move a heavy box from the top of one table to the top of
another table of the same height on the other side of the room. What is the
minimum amount of work you must do on the box to accomplish the move?. Explain.
1.3. True of false: A person on a Ferris wheel is moving in a circle at
constant speed. Thus, no force is doing work on the person.
1.4. By what factor does the kinetic energy of a car change when its
speed is doubled?
1.5. A particle moves in a circle at a constant speed. Only one of the
forces acting on the particle is in the centripetal direction. Does the net
force on the particle do work on it?.
1.6. An object initially has kinetic energy K. The object then moves in
the opposite direction with three times its initial speed. What is the kinetic
energy now? (a) K; (b) 3K; (c) -3K; (d) 9K; (e) - 9K.
1.7. How does the work required to stretch a spring 2 cm from its
natural length compare with that required to stretch it 1 cm from its natural
length?
1.8. Two stones are thrown with the same initial speed at the same
instant from the roof of a building. One stone is thrown at an angle of 30º
above the horizontal, the other is thrown horizontally. (Neglect air
resistance). Which statement below is true?
(a) The stones strike the ground at the same time and with equal speeds.
(b) The stones strike the ground at the same time with different speeds.
(c) The stones strike the ground at different times with equal speeds.
(d) The stones strike the ground at different times with different
speeds.
1.9. True of false:
(a) The total energy of a system cannot change.
(b) When you jump into the air, the floor does work on you increasing
your mechanical energy.
1.10. If a rock is attached to a massless, rigid
rod and swung in a vertical circle at a constant speed, it will not have a
constant total energy, as the kinetic energy of the rock will be constant, but
the potential energy will be continually changing. Is any total work being done
on the rock? Does the rod exert a tangential force on the rock?.
2. NUMERICAL PROBLEMS
2.1. A roller coaster car of mass 1500 kg starts at a distance H = 23 m
above the bottom of a loop 15 m in diameter. If friction is negligible, the downward
for the rails on the car when it is upside down at the top of the loop is: (a)
4.6x104 N, ((b) 3.1x104N,(c) 1.7x104N, (d) 980
N, (e) 1.6x103 N.
2.2. A single-car roller coaster pushes off, and on the first section of
track, descends a 5-m-deep valley, then climbs to the top of a hill that is 9.5
m above the valley floor. (a) What is the minimum initial speed required to
carry the coaster beyond the first hill?. Assume that
the track is frictionless. (b) Can be affect this speed by changing the depth
of the valley to make the coaster pick up more speed at the bottom?.
2.3. A stone is thrown upward at an angle of 53º above the horizontal.
Its maximum height during the trajectory is 24 m. What was the stone´s initial speed?
2.4. A baseball of mass 0.17 kg is thrown from the roof of a building
12 m above the ground. Its initial velocity is 30 m/s at an angle of 40º above
the horizontal. (a) What is the maximum height the ball reaches? (b) What is
the work done by gravity as the ball moves from the roof to its maximum height?
(c) What is the speed of the ball as it strikes the ground?.
2.5. An 80-cm-long pendulum with a 0.6-kg bob is released from rest at
initial angle q0 with
the vertical. At the bottom of the swing, the speed of the bob is 2.8 m/s. (a) What was the initial angle of the pendulum? (b) What angle
does the pendulum make with the vertical when the speed of the bob is 1.4 m/s?.
2.6. A child of mass 40 kg goes down an 8.0-m-long slide inclined at 30º
with the horizontal. The coefficient of kinetic friction between the child and
the slide is 0.35. If the child starts from rest at the top of the slide, how
fast is she travelling when she reaches the bottom?
2.7. A sled is coasting on a horizontal snow-covered surface with an
initial speed of 4 m/s. If the coefficient of friction between the sled and the
snow is 0.14, how far will the sled go before coming to rest?
MORE PROBLEMS
1. a)
A 2000 kg car is travellig 50 miles per hour. Find
the kinetic energy in Joules (1 mi= 1609 m). B) The same car is lifted
vertically upward and then dropped from rest. Find the height from which it is
dropped if it strikes the ground at 50 miles per hour (neglect air resistance).
Sol. 4.99. 105 J; 25.5 m.
2. An object of mass 1 kg
travelling at 5.0 m/s enters a region of ice where the coefficient of kinetic
friction is 0.10. Use the work-energy theorem tto
find the distance the object travels before coming to rest.
Sol. 13 m.
3. A 30 kg child enters the
final section of a waterslide travelling at 2.0 m/s. The final section is 5.0 m
long and has a vertical drop of 3.0 m. The force of friction opposing the child´s motion is 50 N. Find a) the loss of potential
energy, b) the work done by friction in the final section and c) the child´s velocity at the ende of
the section (using energy considerations).
Sol. -882 J; -250 J; 6.8 m/s.
4. A 2.0 kg wood block is on a
level board and held against a spring of spring constant k=100 N/m which has
been compressed 0.1 m. The block is released and pushed horizontally across the
board. The coefficient of friction between the block and the board is mk= 0.20. Find
a) the velocity of the block just as it leaves the spring and b) the distance
the block travel after it leaves the spring.
Sol. 0.33
m/s; 0.028 m.
5. Aman
pushes a 100 kg box across a level floor at a constant speed of 2.0 m/s for 10
s. If the coefficient of friction between the box and the floor is mk= 0.20, find
the average power output by the man.
Sol. 392 W.
6. An automobile of mass 1200
kg has a speed of 30 m/s on a horizontal road when the engine is developing
37300 Watts (50.0 horsepower). What is the speed, with the same power output,
if the automobile now climbs a hill inclined at 30o?
7. In the early 1980's, the
total consumption of electrical energy in the
a) What was the average rate
of energy consumption in watts? kilowatts?
b) If the population of the
c) If the sun transfers energy
to the earth by radiation at a rate of 1.4 kW per square meter of surface, how
great an area would be required to collect the energy cited above?
8. The human heart is a
powerful and reliable pump. Each 24 hour day, it takes in and discharges over
7500 liters of blood. If the work done by the heart
is equal to the work required to lift this amount of blood a height equal to
the average American female (1.63 m), and if the density of blood is the same
as that of water,
a) how
much work does the heart do in a day?
b) what
is the power output in watts? horsepower?
9. A spring has a force
constant (k) of 22.0 N/m. If it oscillates with an amplitude
(A) of 87.0 mm, what is the mechanical energy of the spring?
10. A block with a mass of
0.47 kg is on a harmonic spring whose force constant (k) is 23.0 N/m. While in
motion, its mechanical energy is 0.025 J. What is the amplitude and maximum
speed of the block during the oscillation (ignore the mass of the spring).
11. A common flea (Pulex irritans, Arthropoda: Siphonoptera) has a
mass of 200.0 mg. It can jump approximately 20 cm straight up. Ignoring air
resistance, determine the following for the flea:
a) weight
b) initial
velocity
c) time
of flight
d) initial
momentum
e) initial
kinetic energy
f) maximum
gravitational potential energy
g) its
kinetic and potential energy half way up
h)What
is the mechanical energy of the falling flying flea in flight five cm from the
floor?
12. The speed limit along a
portion of
13. Use logical statements to
explain the following:
a) A woman lets go of her
brief case in an elevator but it does not fall to the floor.
b) A spring with a force
constant of 20 N/m is cut in half. What is its force constant after the cut?
c) A pendulum clock is placed
on an elevator. How does its time period change as the elevator rises?
14. A 200 kg hammer of a pile
driver is lifted 10.0 m.
a) Find the potential energy
of the driver at its maximum height.
b) Find the potential energy
of the driver at 5.0 m.
c) Calculate the velocity of
the driver as it hits the ground when dropped from 10.0 m using the Work-Energy
Theorem.
d) Calculate the velocity of
the driver as it hits the ground when dropped from 5.0 m using the Work-Energy
Theorem.
e) Verify your answers in c
and d using the kinematic equation v2 =
2gs.
15. A 1600 kg car travels at a
speed of 12.0 m/s.
a) Find its kinetic energy.
b) If its speed drops to 6.0
m/s, how does its kinetic energy compare with your answer in (a)?
c) How does doubling the speed
of a car affect its braking distance. Assume the same
work is done by the brakes in each case.
16. A 20.0 kg mass is on the
edge of a 100.0 m cliff.
a) What is its potential
energy?
b) What is its potential
energy 50.0 m above the ground?
c) What is its potential
energy 25.0 m above the ground?
d) Using the W-E Theorem,
calculate its velocity 40 m above the ground.
e) What becomes of the energy
after hitting the ground?
17. A steel ball has a mass of
40 g and rolls along a smooth level surface at 0.42 m/s. Find
its kinetic energy.
18. A nitrogen gas molecule (N2)
has a speed of 450 m/s. Find its kinetic energy if its mass is 4.54 x 10-26
kg.
19. An electron's energy is
measured at 2.81 x 10-14 J when it is traveling
at 2.5 x 108 m/s. Determine the mass of an electron.
20. A 7.2 kg bowling ball
rolls down a frictionless channel from a height of 0.7 m. Determine
its velocity at the bottom.
21. In the sketch below, a
ball is shown on a frictionless path.
a) to
what point does the ball rise on the opposite incline?
b) at
what point on the sketch is the speed maximum?
c) at
what point is the speed zero?
d) along
which path is speed constant?
22. What types of energy do
the following items have? (potential (P.E.), kinetic
(K.E.), gravitational potential (G.P.E.); you may use more than one.
a) moving
ping pong ball
b) AA battery on the top shelf
c) wind
up pendulum clock
d) basketball
on the way to the net
e) compressed
spring in a ball point pen
23. A BIC Clic®
pen requires a force of 1.4 N to push the ink cartridge a distance of 0.7 cm.
a) What is the force constant
of the spring?
b) What energy has been added
to the spring?
c) If the spring is released,
what is the velocity of the pen's clicker upon release if its mass is 0.5 g?
24. A child is on a swing. Her
mom pushes her to a height of 1.5 m above the starting position. Determine the
child's maximum velocity during the swing.