Work and Energy

AP Physics BWORK AND ENERGY

WORK

• Work is the transfer of energy to or from a system by the application of forces exerted on the system by the environment.

• Work is done on a system by forces from outside the system (external forces).

• Internal forces—forces between objects within the system—cause energy transformations within the system, but don’t change the system’s total energy.

• In order for energy to be transferred as work, the system must undergo a displacement. In other words, the system must move during the time the force is applied.

WORK

• Work done by a constant force F in the direction of a displacement ∆r is equal to the product of these two:

• The unit of work is N∙m. This unit is so important that it has been given its own name, the joule.

• The joule is the unit of all forms of energy.• Work is a scalar quantity.

WORK

Sarah pushes a heavy crate 3.0 m along the floor at constant speed. She pushes with a constant horizontal force of magnitude 70 N. How much work does Sarah do on the crate?

EXAMPLE 1WORK DONE IN PUSHING A CRATE

A force does the greatest possible amount of work on an object when the force points in the same direction as the object’s displacement.

Less work is done when the force acts at an angle to the displacement.

Work done by a constant force F at an angle to the displacement is equal to:

Quantities of F and ∆r are always positive, so the sign of W is determined entirely by the angle between the force and the displacement.

FORCE AT AN ANGLE TO THE DISPLACEMENT

A strap inclined upward at a 45° angle pulls a suitcase through the airport. The tension in the strap is 20 N. How much work does the tension do if the suitcase is pulled 100 m at a constant speed?

We can use the work equation, with F = T, to find that the tension does work:

EXAMPLE 2WORK DONE IN PULLING A

SUITCASE

KINETIC ENERGY

Consider a car being pulled by a tow rope. The rope pulls with a constant force F while the car undergoes displacement ∆r, so the force does work W = F∆r on the car.

Ignoring friction and drag, word done by F is transferred entirely into the car’s energy of motion—its kinetic energy. The change in the car’s kinetic energy is equal to the work done.

Using kinematics, we can find another expression for the work done.

These two combined result in the work-energy theorem.

KINETIC ENERGY

A two-man bobsled has a mass of 390 kg. Starting from rest, the two racers push the sled for the first 50 m with a net force of 270 N. Neglecting friction, what is the sled’s speed at the end of the 50 m?

EXAMPLE 3SPEED OF A BOBSLED AFTER

PUSHING

From the work-energy theorem, the change in the sled’s kinetic energy is ∆K = K f – Ko = W.

The sled’s final kinetic energy is thus

Substituting the expressions for kinetic energy and work gives:

Vo = 0, so the this equation simplifies to

This can be solved for final velocity to get

EXAMPLE 3SPEED OF A BOBSLED AFTER

PUSHING

POTENTIAL ENERGY

When two or more objects in a system interact. It is sometimes possible to store energy in the system in a way that the energy can be easily recovered. This sort of stored energy is called potential energy, since it has the potential to be converted into other forms of energy.

Forces due to gravity and springs are special in that they allow for this storage of energy. Other interaction forces do not.

Interaction forces that can store useful energy are called conservative forces.

Forces that do not store energy are called nonconservative.

POTENTIAL ENERGY

Think of lifting a book at constant speed. Constant speed means the kinetic energy doesn’t change. Thus, the work done on the book goes entirely into increasing the potential energy. Representing potential energy with variable U:

Because the above equation can be written as

The work done is , where is the vertical distance that the book is lifted, or the height, h.

, so , and so

The higher the object is lifted, the greater the gravitational potential energy.

GRAVITATIONAL POTENTIAL ENERGY

Because the change in potential energy is entirely dependent on the change in height, we are free to choose a reference level where we define to be zero, so potential energy is simplified as

An important conclusion to take from this equation is that gravitational potential depends only on the height of the object and not the object’s horizontal position, or not on the path the object took to get to that position.

GRAVITATIONAL POTENTIAL ENERGY

In the Empire State Building Run-Up, competitors race up the 1576 steps of the Empire State Building, climbing a total vertical distance of 320 m. How much gravitational potential energy does a 70 kg racer gain during this race?

We choose at the ground floor of the building, so, at the top, the racer’s gravitational potential energy is

EXAMPLE 4RACING UP A SKYSCRAPER

Energy can also be stored in a compressed or extended spring as elastic potential energy.

A force is required to compress the spring. This force does work on the spring, transferring energy to the spring. So, how much elastic potential energy is stored in the spring can be found by calculating the amount of work needed to compress the spring.

ELASTIC POTENTIAL ENERGY

When no forces act on a spring to compress or extend it, it will relax to its equilibrium length.

If we compress the spring by a displacement with a force , by Newton’s third law there is also an equal but opposite force that tries to push the spring back to equilibrium.

This force, called the spring force, , is proportional to the displacement, , of the end of the spring.

The minus sign is because the spring force always points in the opposite direction of the displacement.

is called the spring constant. Spring constants are measured in N/m.

The spring constant is a property that characterizes a spring. Springs with a large spring constant are more difficult to compress or extend than those with a small spring constant.

HOOKE’S LAW

When dealing with potential energy, the reference level is the point where potential energy is equal to zero. This is the point where a spring is at equilibrium length, so any displacement is equal to the final position .

The spring force is , so the applied force is .As the applied force pushes the end of the spring

from its equilibrium position to a final position , the applied force increases from to .

This is not a constant force—more compression requires a larger applied force—so we can’t use to find the work done.

SPRING POTENTIAL ENERGY

An average force can be used to calculate the work done on a spring.

Because the force varies in magnitude from to , the average force used to compress the spring is .

Thus the work done by the applied force is

This work is stored as potential energy in the spring, so elastic potential energy for a spring is

SPRING POTENTIAL ENERGY

An archer pulls back the string on her bow to a distance of 70 cm from its equilibrium position. To hold the string at this position takes a force of 140 N. How much elastic potential energy is stored in the bow?

A bow is an elastic material, so we will model it as obeying Hooke’s law, , where is the distance the string is pulled back.

We can use the force required to hold the string, and the distance it is pulled back, to find the bow’s spring constant .

Then, we can use the equation for elastic potential energy.

EXAMPLE 5PULLING BACK ON A BOW

From Hooke’s law, the spring constant is

Then the elastic potential energy of the flexed bow is

EXAMPLE 5PULLING BACK ON A BOW

Thermal energy is related to the microscopic motion of the molecules of an object, and in turn is related to the kinetic and elastic potential energy of said molecules.

Increasing an object’s thermal energy also increases its temperature.

Imagine a box pulled across a floor at a constant speed. The force required to pull the box does work on the system, which therefore transfers energy into the system.

Because the speed is constant, there is no change in kinetic energy.

Because the box’s height is constant, there is no change in gravitational potential energy.

The increased energy must be in the form of thermal energy.

THERMAL ENERGY

Increase in thermal energy is a general feature of any system where friction between sliding objects is present. Change in thermal energy, , is related to friction, , and displacement, , as:

This equation does not only apply to object’s being acted on by an external forces. For example, thermal energy can be created by a moving object sliding to a halt on a rough surface.

THERMAL ENERGY

A 0.30 kg block of wood is rubbed back and forth against a wood table 30 times in each direction. The block is moved 8.0 cm during each stroke and pressed against the table with a force of 22 N. How much thermal energy is created in the process? The coefficient of friction between two wooden surfaces is μ = 0.20.

Use NSL in the y-direction to calculate normal force.Use to calculate friction force.Use to calculate thermal energy created.

EXAMPLE 6THERMAL ENERGY BY RUBBING

The block is not accelerating in the y-direction, so Newton’s second law gives

Friction force is then

Total displacement of the block is

So thermal energy created is

EXAMPLE 6THERMAL ENERGY BY RUBBING

LAW OF CONSERVATION OF

ENERGY

Recall that work is the transfer of energy to or from a system by a force acting on the system. The total energy of a system changes by the amount of work done on it.

Like conservation of momentum, if we define a system that has no external forces acting on it, then the total energy of this isolated system is conserved, and change in energy is zero.

This applies to every form of energy (kinetic, chemical, nuclear, etc.), but we often only concern ourselves with the kinetic and potential energy of a system because these are the energies associated with the motion and position of an object. Together these energies are referred to as the mechanical energy of the system.

ENERGY TRANSFER AND CONSERVATION

If the system is isolated and there’s no friction or friction is negligible, the mechanical energy is conserved:

If the system is isolated but there is friction within the system, the total energy is conserved:

CONSERVATION OF MECHANICAL ENERGY

A the county fair, Katie tries her hand at the ring-the-bell attraction. She swings the mallet hard enough to give the ball an initial upward speed of 8.0 m/s. Will the ball ring the bell, 3.0 m from the bottom? Assume the track on which the ball moves is frictionless.

Identify the mechanical energies involved in this system, and set the sum of the initial values of these energies equal to the sum of the final values of these energies.

Simplify this expression using values and/or the energies’ corresponding equations.

Solve the expression for height, .

EXAMPLE 7HITTING THE BELL

Once the ball is in motion, only kinetic energy and gravitational potential energy are involved, so conservation of mechanical energy is

Substituting the equations for kinetic and gravitational potential energy gives

The ball starts at , and at its highest point, , so the above expression simplifies to

Solving this for height gives

EXAMPLE 7HITTING THE BELL

Still at the county fair, Katie tries the water slide. The starting point is 9.0 m above the ground. She pushes off with an initial speed of 2.0 m/s. If the slide is frictionless, how fast will Katie be traveling at the bottom?

EXAMPLE 8SPEED AT THE BOTTOM OF A SLIDE

Conservation of mechanical energy gives

Or

Taking ,

Which we can solve for

EXAMPLE 8SPEED AT THE BOTTOM OF A SLIDE

A spring-loaded toy gun is used to launch a 10 g plastic ball. The spring, which has a spring constant of 10 N/m, is compressed by 10 cm as the ball is pushed into the barrel. When the trigger is pulled, the spring is released and shoots the ball back out. What is the ball’s speed as it leaves the barrel? Assume that friction is negligible, and the spring is massless.

EXAMPLE 9SPEED OF A SPRING-LAUNCHED

BALL

Energy conservation equation is . We can use elastic potential of the spring to write this as

We know that and , so this simplifies to

Which we can solve for to give

EXAMPLE 9SPEED OF A SPRING-LAUNCHED

BALL

POWER

In many situations, it is necessary to know how quickly energy is transformed or transferred. This implies a rate of transformation of energy.

The amount of energy transformed over a time interval is called the power and is defined as

The unit of power is the watt, which is defined as 1 watt = 1 W = 1 J/s.

Recall that so a more conventional way of representing power is

One other useful was of representing the power formula is

ENERGY, WORK AND TIME

Your 1500 kg car is behind a truck traveling at 60 mph (27 m/s). To pass it, you speed up to 75 mph (34 m/s) in 6.0 s. What engine power is required to do this?

Calculate the work required to speed up the car to the desired final velocity. Recall that .

Calculate power using work and time.

EXAMPLE 10POWER TO PASS A TRUCK

To transform this amount of energy in 6 s, the power required is

EXAMPLE 10POWER TO PASS A TRUCK

A truck’s brakes can overheat and fail while descending mountain highways, leading to an extremely dangerous runaway truck. Some highways have runaway-truck ramps to safely bring out-of-control trucks to a stop. These uphill ramps are covered with a deep bed of gravel. The uphill slope and the large coefficient of rolling friction as the tires sink into the gravel bring the truck to a safe halt.

EXAMPLE 11STOPPING A RUNAWAY TRUCK

A 22,000 kg truck heading down a 3.5° slope at 20 m/s suddenly has its brakes fail. Fortunately, there’s a runaway-truck ramp 600 m ahead. The ramp slopes upward at an angle of 10°, and the coefficient of rolling friction between the truck’s tires and the loose gravel is . Ignore air resistance and rolling friction as the truck rolls down the highway.

a. Use conservation of energy to find how far along the ramp the truck travels before stopping.

b. By how much does the thermal energy of the truck and ramp increase as the truck stops?

EXAMPLE 11STOPPING A RUNAWAY TRUCK