Lesson 6: Work and Energy

Question 1

What is the work done by a force F which acts on an object as it moves through a distance d?

Answer 1

W = F · d

Were Work is the dot product of the force and displacement vectors or:

W = F*d cos(θ)

Where θ is the angle between the force and the direction of motion.

Since work is proportional to cos(θ), only the component of force parallel to the motion contributes to the work; any forces perpendicular to motion do no work since cos(90)=0.

Question 2

What are the units of work, as defined in physics?

Answer 2

Work has units of Joules (J).

Remember: work is defined as F * d cos(θ), which has units of N * m, or kg * m2/s2, which is identical to the units of Joules.

Question 3

What is the sign of work done on an object if it begins at rest and an applied force accelerates in the same direction as the displacement of the object?

Answer 3

The work done on the object by the force is positive (+).

By definition, work done by a force that leads to a change in distance (hence an increase in speed) is positive work if it accelerates in the same direction as the displacement of the object.

If the force has a component in the direction opposite to the displacement, the force does negative work.

Question 4

What is the sign of work done on an object, if it begins at speed v and an applied force decelerates it to rest?

Answer 4

The work done on the object by the force is negative (-).

By definition, work done by a force (such as friction) that leads to a decrease in speed is negative work.

Question 5

What kind of work, positive or negative, can a person do by pushing on a box?

Answer 5

A person can do either positive or negative work by pushing on a box.

If the person pushes on the box in the same direction that it moves, then positive work is done. If the person pushes in the opposite direction of the moving box, then negative work is done.

Question 6

What kind of work (positive, negative, or both) can frictional forces do?

Answer 6

Frictional forces can only do negative work.

By definition, frictional forces are always in the opposite direction of an object’s motion. Hence, they can only slow the object down, and only do negative work.

Question 7

What is the work done by gravity as an object of mass m moves from the ground to a height h?

Answer 7

W(Gravity) = -mgh

Remember, W = F * d cos(θ). In this case, as an object moves straight up, θ = 0º and cos(θ) = 1. So, W = F * d. The force of gravity is simply -mg, and the distance is h, so the total work done is -mgh.

Work is negative in this case, because gravity works against the object being moved.

Question 8

An object at rest is moved from the ground to a height h/2, then to rest at a height h. How much work does gravity do during this process?

Answer 8

W = -mgh

The work done by gravity while the object moves to h/2 is -mg(h/2). The work done while the object is moving from h/2 to h is also -mg(h/2). Work is negative in this case, because gravity works against the object being moved.

Notice that this is identical to the work done if the object is moved directly to h; the work done by gravity is path-independent.

Question 9

Define:

power, as used in physics

Answer 9

Power is a measure of the rate of energy flow.

Power is defined as energy divided by time:

P = E/t

The units of power are Watts, where 1 W = 1 J/s.

Question 10

What is the power flowing through a wire, if 1,000 J of energy flow through in 0.1 s.

Answer 10

P = 10,000 W = 10 kW

Power is energy divided by time;

1000 J / 0.1 s = 10,000 W

Note that kW is a commonly-used unit

Question 11

Define:

the kinetic energy of an object

Answer 11

An object’s kinetic energy is the energy resulting from its motion.

Kinetic energy is defined as:

KE = ½mv^2

where m is the object’s mass and v is its speed. The units of kinetic energy are Joules, just like all other forms of energy.

Question 12

If objects 1 and 2 are moving at the same speed, but object 2 has twice the mass of object 1, how do their kinetic energies compare?

Answer 12

KE2 = 2*KE1

Kinetic energy is defined as ½mv^2, so it is directly proportional to the mass. If the objects’ speeds are the same, kinetic energy will increase proportionally with the mass.

Question 13

If objects 1 and 2 are the same mass, but object 2 is moving at twice the speed of object 1, how do their kinetic energies compare?

Answer 13

KE2 = 4*KE1

Kinetic energy is defined as ½mv^2, so it is directly proportional to the square of the velocity. If the objects’ masses are the same, kinetic energy will increase with the square of the velocity.

Question 14

Define:

gravitational potential energy

Answer 14

Gravitational potential energy is energy an object possesses because of its position in a gravitational field.

Gravitational potential energy is defined as:
Ugr = mgh

Where m is mass in kg, g is the gravitational acceleration in m/s^2 and h is the height in the gravitational field above some datum.

The units of gravitational potential energy are Joules, just like all other forms of energy.

Question 15

Define:

the work-energy theorem

Answer 15

The work-energy theorem states that the net work done on an object by all the forces acting on it equals the change in the object’s kinetic energy.

Wnet = ΔKE

Question 16

An object is at rest, and a person pushes on it, doing 50 J of work that convert to motion. What is the object’s final kinetic energy? (assume no dissipative forces)

Answer 16

The object’s final KE is 50 J.

According to the work-energy theorem, the net work done on the object results in a change in its kinetic energy. Since the person is exerting the only force on the object which causes motion, the work done equals the change in kinetic energy.

Question 17

An object is moving with 100 J of kinetic energy, and a frictional force acts on it, doing 50 J of work. What is the object’s final kinetic energy?

Answer 17

The object’s final KE is 50 J.

According to the work-energy theorem, the net work done on the object results in a change in its kinetic energy. Since the frictional force is the only force affecting motion, the work done equals the change in kinetic energy. Frictional forces can only do negative work, so the final kinetic energy is lowered.

Question 18

An object with mass m falls through a distance h. What is its final kinetic energy? (assume no dissipative forces)

Answer 18

The object’s final KE is mgh.

According to the work-energy theorem, the net work done on the object results in a change in its kinetic energy. Since gravity is the only force acting on the object, the change in kinetic energy simply equals the work done. Work = F * d, or, in this case, mg * h. You might also recognize this as the gravitational potential energy of the object before it fell.

Question 19

How does the gravitational potential energy of a mass change, if its height above the surface of the Earth doubles?

Answer 19

The gravitational potential energy of the mass doubles.

Since gravitational potential energy of an object near Earth’s surface Ugr = mgh is proportional to height above the ground, changing the height changes the energy by the same amount.

Question 20

What is the potential energy of gravitational attraction between two objects of masses m1 and m2?

Answer 20

Ugr = -Gm1m2/r

G is the universal gravitational constant, and r is the distance between the centers of mass for the two objects.

This expression applies in all scenarios. Hence, is different from Ugr=mgh, which only approximates that near the Earth’s surface, where gravitational acceleration is roughly constant (equal to g, ~10 m/s^2).

Question 21

How does the gravitational potential energy between two masses change, if the distance between the two masses doubles?

Answer 21

The gravitational potential energy between the masses increases.

Be careful of signs! The general expression for gravitational potential energy, Ugr = -Gm1m2/r, is always negative. Increasing r makes the fraction smaller; in this case, less negative, reflecting an increase in potential energy.

Question 22

What is the force on a mass m, attached to a stretched spring, at a distance x from equilibrium?

Answer 22

F = -kx

Where k is the spring constant (or force constant) of the spring (in N/m) and x is the distance from equilibrium. Notice that the negative sign means force will always act in the opposite direction of net displacement.

Question 23

If spring 1 has twice the value of k that spring 2 has, which spring will need to be moved the greater distance in order to create the same force?

Answer 23

Spring 2 (the lower k) will need to be displaced further.

Since F=-kx and spring 2 has 1/2 the k value of spring 1, spring 2 will need to move twice the distance in order to create the same value of F.

Question 24

Define:

the potential energy of a spring which has been stretched/compressed

Answer 24

A spring’s potential energy Usp is defined as the work needed to stretch or compress the spring from its equilibrium length to the current length.

The spring’s equilibrium length will be defined as
x = 0.

Question 25

What is a spring’s potential energy, if a mass m is attached and the spring is stretched to a distance x from equilibrium?

Answer 25

Usp = ½kx^2

Where x is the distance from equilibrium and k is the spring constant. There is no difference between sign for compression and expansion, since x^2 will always be positive.

Question 26

How does the potential energy of a spring change, if its extension from equilibrium doubles?

Answer 26

The potential energy of the spring increases by a factor of 4.

Since the potential energy of the spring Usp = ½kx^2 is proportional to the square of the displacement of the spring, stretching the spring increases the potential energy by the square of the proportional length change.

Question 27

Define:

the total mechanical energy of a physical system

Answer 27

The total mechanical energy of a physical system is the sum of the kinetic and potential energies of all the objects which make up the system.

ME(total) = KE(total) + U(total)

Question 28

What specific energies constitute the total mechanical energy of a mass oscillating on a horizontal spring?

Answer 28

The kinetic energy from the mass’s velocity, and the potential energy due to the extension or compression of the spring from equilibrium.

Question 29

Define:

the principle of conservation of energy

Answer 29

The principle of conservation of energy states that, in the absence of dissipative forces, a system’s total mechanical energy is constant.

Dissipative forces such as friction or air resistance will decrease a system’s total mechanical energy.

Question 30

Define:

a conservative force in physics

Answer 30

A conservative force is one which does not dissipate mechanical energy.

When only conservative forces are acting on a system, the system’s total mechanical energy will stay constant.

Question 31

Define:

a nonconservative force in physics

Answer 31

A nonconservative force is one which dissipates mechanical energy.

When nonconservative forces are acting on a system, the system’s total mechanical energy will decrease.

Question 32

A box is pushed at some speed across the floor by a force and then released. How does the total mechanical energy of the box at the point of release compare to the box at a later time?

Answer 32

The total mechanical energy later will be less than when released.

The box only has kinetic energy due to velocity, but the dissipative force of friction acts between the box and the floor. Friction inhibits motion, thus kinetic energy, and total mechanical energy, will be less the longer friction acts.