NPS 210- Physics for Engineers
WORK , ENERGY & POWER
WORK (W) = is accomplished by the action of a force when it acts on an object and the object moves through a distance
= a scalar quantity
= is equal to the product of the magnitude of the displacement (s) and the component of the force parallel to the
displacement
In equation:
→ →
W =F • s =F ¿ s
or W=( F cosθ ) s work done by a constant or specific force
or W=Fs cos θ
If F and s are in the same direction, = 0, and cos 0 = 1, then W = Fs. Work is positive.
If F and s are in opposite direction, = 180, and cos 180 = -1, then W = -Fs. Work is negative.
If F and s are perpendicular to each other, = 90, and cos 90 = 0, then W =0. No work is done.
If two or more forces act on the object, the work done on the object is:
Wtot = WF + Wf + WT + WW + ….. or Wtot = F s
Units of work: Equivalents:
In mks system: joule (J) = Newton meter 1 J = 107 erg = 0.7376 ft lb
In cgs system: erg = dyne centimeter
In fps system: foot-pound (ft-lb) = pound feet
The work done by a variable force in moving an object between two points is equal to the area under the F ll vs. distance
curve between these two points.
Sample Problems:
1. a) Steve exerts a steady force of magnitude 210 N on the stalled car as he pushes it a distance of 18 m. The car also has a
flat tire, so to make the car straight Steve must push at an angle of 30 to the direction of motion. How much work does
Steve do? b) In a helpful mood, Steve pushes a second stalled car with a steady force ⃗ F =( 160 N ) i^ −(40 N ) ^
j . The
^
displacement of the car is ⃗s= (14 m ) i+(11 m) ^
j . How much work does Steve do in this case?
2. You push your physics 1.5 m along a horizontal table-top with a horizontal push of 2.40 N while the opposing force of
friction is 0.600 N. How much work does each of the following forces do on the book: a) your 2.40 N push, b) the friction
force, c) the normal force from the table top, and d) gravity? What is the net work done on the book?
3. A 50-kg crate is pulled 40 m along a horizontal floor by a constant force exerted by a person, Fp = 100 N, which acts at a 37
angle. The floor is rough and exerts a frictional force f = 50 N. Determine the work done by each force acting on the crate
and the net work done on the crate.
4. (a) Determine the work a hiker must do on a 15.0 kg backpack to carry it up the hill of height 10.0 m. Determine also (b) the
work done by gravity on the backpack, and (c) the net work done on the backpack. Assume the motion is at constant
velocity.
5. Calculate the work done by a woman (a) when slowly lifting a 15-kg suitcase 0.80 m upward, (b) when lowering it 0.80 m,
and (c) when holding it at rest.
6. A woman jumps from a high wall and lands on soft sand. The sand exerts an average upward force of 50,000 N while
stopping her. The woman stops after sinking 5 cm into the sand. How much work is done by the force of the sand on her?
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ENERGY (E) = the capacity to do work
= a scalar quantity
Types:
1. kinetic energy (K) = is the energy possessed by an object because of its motion
1
K= m v 2
2 translational kinetic energy
ΔK =K− K o
1 1
ΔK = mv 2 − mv 2
or 2 2 o where v o & v are the initial & final velocities
WORK-ENERGY THEOREM
“ The net work done on an object is equal to the change in its kinetic energy”.
W tot =ΔK
2. potential energy (U) = is the energy an object has by virtue of its position and configuration
a) gravitational potential energy (Ug) = the potential energy due to gravitational forces
U g =mgy ΔU =U −U o
ΔU =mgy−mgy o
b) elastic potential energy (Us ) = potential energy of elastic materials
1
U s= k x2 F s=−kx
2 Spring equation or Hooke’s Law:
1 1
ΔU = kx 2 − kx 2
2 2 o
nonconservative forces = are forces such as friction, tension, & actual force, for which the work done depends on
whether the path taken is straight, or is curves or zigzag.
“ The work done by the nonconservative(NC) forces acting on an object is equal to the total change in kinetic energy and potential
energy”.
W tot =W NC =ΔK + ΔU
conservative forces = are forces such as gravity, elastic, & electric, for which the work done does not depend on the
path taken but only on the initial and final positions.
If gravity is the only force that acts on the object, then W tot =W grav =−∆ U and ∆ K =−∆ U
CONSERVATION OF MECHANICAL ENERGY
“ If only conservative forces are acting on an object, the total mechanical energy of a system neither increases not decreases
in any process. It stays constant –it is conserved”.
ΔK + ΔU =0 1 2 1 2
m v 2− m v 1+ mg y 2−mg y 1=0
2 2
TOTAL MECHANICAL ENERGY: E=K +U = constant (if only gravity does work)
Sample Problems:
1. A 0.10-kg stone is thrown from the edge of an ocean cliff with an initial speed of 21 m/s. When it strikes the water below, it
is traveling at 45 m/s. What is the change in kinetic energy of the stone?
2. A 400-kg roller-coaster car starts from 45 m above the ground and is 4 m above the ground at the end of the ride. Calculate
the change in gravitational potential energy of the car.
3. A children’s slide is 20 ft long and makes an angle 30 with the horizontal. If the coefficient of sliding friction is 0.05, and the
child starts from rest at the top, with what speed does he reach the bottom?
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4. If the original height of the stone is 3.0 m, calculate the stone’s speed when it has fallen to 1.0 m above the ground.
5. In one day, a 75-kg mountain climber ascends from the 1500-m level on a vertical cliff to the top at 2400 m. The next day,
she descends from the top to the base of the cliff, which is at an elevation of 1350 m. What is her change in gravitational
potential energy a) on the first day and b) on the second day?
6. You throw a 0.145-kg baseball straight up, giving it an initial velocity of magnitude 20.0 m/s. Find how high does it goes,
ignoring air resistance.
7. A force of 800 N stretches a certain spring a distance of 0.200 m. a) What is the potential energy of the spring when it is
stretched 0.200 m? b) What is its potential energy when it is compressed 5.00 cm?
8. A 2,000-kg elevator with broken cables in a test rig is falling at 4.00 m/s when it contacts a cushioning spring at the bottom
of the shaft. The spring is intended to stop the elevator, compressing 2.00 m as it does so. During the motion a safety
clamp applies a constant 17,000-N frictional force to the elevator. What is the necessary force constant k for the spring?
POWER (P) = the rate at which work is done
= the rate at which energy is transformed
W Fd
P= P=
t or t or P=F v
Units for power:
watts (W) = joules per second
kilowatts (kW) = 1000 W = 1.34 horsepower
horsepower (hp) = 550 ft lb per second = 746 watts
Therefore, the total work done in 1 hour, when the rate of doing work is kilowatt, is 1 kilowatt-hour (kWh). When
the rate of doing work is 1 horsepower, the total work done in 1 hour is 1 horsepower-hour (hph). Thus, kilowatt-hour and
horsepower-hour are units of energy.
Sample Problems:
1. A crane lifts a 300-kg load at constant speed a vertical distance of 30 m in 10 seconds. Calculate the rate at which the crane
is doing work on the load.
2. A 70-kg jogger runs up a long flight of stairs in 4.0 seconds. The vertical height of the stairs is 4.5 m. Estimate the jogger’s
power output in watts and horsepower. How much energy did this require?
3. (a) Calculate the power needed from the motor of an elevator for it to lift a 10000-kg mass a distance of 20 m in 5.0
seconds.
(b) Calculate the power output in hp needed for the pump of a fire engine that must lift 30 kg of water a vertical distance of
20 m each second.
4. As part of a charity fund-raising drive a marathon runner with mass 50 kg runs up the stairs to the top of the 443-m tall
building. In order to lift herself to the top in 15.0 minutes, what must be her average power output in watts? In kilowatts?
In horsepower?
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