Work & 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 = Fd cos(θ)

Here, 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 the motion do no work since cos(90)=0.

Question 2

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

Answer 2

Joules (J)

Remember that work is defined as F * d cos(θ), which has units of N * m, or kg * m²/s². These are 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 it to a speed v?

Answer 3

positive (+)

By definition, work done by a force that leads to a change in distance, and hence an increase in speed, is positive 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

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

either positive or negative work

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

Question 6

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

Answer 6

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 expression gives the work done by gravity as an object of mass (m) moves from the ground to a height (h)?

Answer 7

W = 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.

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:

mechanical advantage

Answer 9

It is the multiplication of a force using a mechanical device.

A small force exerted over a large distance is transformed into a larger force over a smaller distance.

On the MCAT, mechanical advantage appears primarily in problems including levers and pulleys.

Question 10

What is the relationship between force and distance in any system which includes mechanical advantage?

Answer 10

F₁d₁ = F₂d₂

In any system exhibiting mechanical advantage, the force exerted and the distance covered are inversely proportional. This occurs because work must be the same in both cases.

Question 11

A force F₁ is exerted a distance d₁ from the fulcrum of a lever. What does the force F₂ at d₂ from the other end equal?

Answer 11

F₂ = F₁d₁ / d₂

In any system exhibiting mechanical advantage, the force exerted and the distance are inversely proportional, F₁d₁ = F₂d₂. Rearranging yields the above relationship.

Question 12

A pulley system is set up that allows a weight m to be lifted using a force of only (1/3)mg. How far must the string on which the force is exerted be pulled in order to move the weight a distance of d?

Answer 12

3d

In any system exhibiting mechanical advantage, the force exerted and the distance are inversely proportional, F₁d₁ = F₂d₂. If one-third of the force is required, the distance must increase by the same proportion.

Question 13

Define:

power

Answer 13

• It 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 14

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

Answer 14

P = 10,000 W = 10 kW

Power is energy divided by time;

1000 / 0.1 = 10,000 W

Note that the kW is a commonly-used unit on the MCAT.

Question 15

Define:

the kinetic energy of an object

Answer 15

• The energy resulting from its motion.
• Kinetic energy is defined as:

KE = ½mv²

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

Question 16

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 16

KE₂ = 2KE₁

Kinetic energy is defined as ½mv², so it is directly proportional to mass. If the objects’ speeds are the same, kinetic energy will increase proportionally with mass.

Question 17

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 17

KE₂ = 4KE₁

Kinetic energy is defined as ½mv², so it is directly proportional to the square of the velocity. If the objects’ masses are the same, kinetic energy will increase with velocity squared.

Question 18

Define:

work-energy theorem

Answer 18

It 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.

W_net = ΔKE

Question 19

An object is initially at rest. A person then pushes on it, doing 50 J of work that convert to motion. What is the object’s final kinetic energy?

Assume no dissipative forces are present.

Answer 19

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 20

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 20

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 21

An object with mass (m) falls from a height (h). What is its final kinetic energy?

Assume no dissipative forces are present.

Answer 21

final KE is equal to 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 22

Define:

gravitational potential energy

Answer 22

An object’s gravitational potential energy (U_gr) is defined as the work that was needed to raise it to its present height above an arbitrary reference point.

On the MCAT, the reference point will always be ground level, unless otherwise stated.

Question 23

What is the gravitational potential energy of an object of mass (m) that is a height (h) above the surface of the Earth?

Answer 23

U_gr = mgh

This expression is identical to the work needed to raise the object that height, and is also equal to the kinetic energy the object will gain if it is allowed to fall freely to the ground.

Question 24

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

Answer 24

The gravitational potential energy of the mass doubles.

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

Question 25

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

Answer 25

U_gr = -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. In that way, it is different from U_gr=mgh, which only approximates that near the Earth’s surface gravitational acceleration is roughly constant (equal to g, ~10 m/s²).

Question 26

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

Answer 26

The gravitational potential energy between the masses increases.

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

Question 27

How can the potential energy of a spring be described?

Answer 27

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

On the MCAT, the spring’s equilibrium length will be defined as x = 0.

Question 28

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 28

U_sp = ½kx²

Here, x is the distance from equilibrium and k is the spring constant. There is no difference related to the sign for compression and expansion, since x² will always be positive.

Question 29

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

Answer 29

It increases by a factor of 4.

Since the potential energy of the spring (U_sp = ½kx²) is proportional to the square of the displacement of the spring, stretching the spring increases the potential energy by the square of the length change.

Question 30

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

Answer 30

F = -kx

Here, 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 31

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

Answer 31

Spring 2 (with 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 to create the same value of F.

Question 32

Define:

total mechanical energy

Answer 32

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.

E_total = KE_total + PE_total

On the MCAT, you will rarely have to track more than one kinetic and one potential energy in any system.

Question 33

What specific energies constitute the total mechanical energy of a cannonball in flight?

Answer 33

The total mechanical energy consists of the kinetic energy from the cannonball’s velocity and the gravitational potential energy due to its height above the ground.

Assume the canonball is not rotating, unless explicitly mentioned. Rotational kinetic energy is rarely tested on the MCAT.

Question 34

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

Answer 34

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

Question 35

What does the principle of conservation of energy state?

Answer 35

It 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 36

At the bottom of an initial hill, a roller coaster car is moving with a speed of v. Ignoring friction, what determines whether it can glide to the top of the next hill?

Answer 36

The car will reach the top of the next hill if its kinetic energy at the bottom K_i is greater than the gravitational potential energy will be at the top of the hill U_f.

Energy conservation states that total energy K+U_gr remains constant. At the bottom of the hill, potential energy is zero. If K_i > U_f, then the initial kinetic energy is sufficient to overcome the force of gravity and climb the hill. If K_i < U_f, the car will not make it to the top of the next hill.

Question 37

Define:

conservative force

Answer 37

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 38

What are some common examples of conservative forces in physics?

Answer 38

The most common conservative forces on the MCAT are:

• gravity
• electrostatic and magnetic forces
• restorative force of a stretched spring

Question 39

A spring with a mass on each end is floating in space, and one mass is pushed towards the other to compress the spring and then released. How does the total mechanical energy of the compressed spring compare to the oscillating spring?

Answer 39

The total mechanical energy will be the same.

The compressed spring has only potential energy of position, and the oscillating spring can only have as much kinetic energy at equilibrium as was already present. Since no dissipative forces exist, and springs are conservative, total mechanical energy will remain constant.

Question 40

Define:

nonconservative force

Answer 40

One which dissipates mechanical energy.

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

Question 41

What are some common examples of nonconservative forces in physics?

Answer 41

The most common nonconservative forces on the MCAT are:

• friction
• air resistance
• fluid viscosity

Question 42

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?

Assume that this process occurs as it would in real life.

Answer 42

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

The box itself only has kinetic energy due to velocity, but the dissipative force of friction acts between the box and the floor. Friction inhibits motion, so kinetic energy will be less the longer friction acts. Total mechnical energy, then, will also be less.