SAT Physics Energy, Work, and Power - Power

SAT Physics Energy, Work, and Power - Power

POWER
Power, P,is the amount of work done over a period of time. It is the rate at which work is done. It is also the rate of energy use or energy generation. It can be calculated using the fol­lowing equation.

Since W=Fd, the above formula can be written as

Distance divided by time should be very familiar to you. It is velocity. Therefore, power can also be expressed as

P = Fv

Power has the units of watts (W). When you look at the top of a lightbulb, it is labeled in watts. A watt is the rate at which the lightbulb uses energy. Another way to express a watt is a joule per second. A 100-watt lightbulb uses 100 joules of energy every second. Power is a scalar quantity. It has magnitude but no direction.

Determining the Power Required to Lift an Object

Determine the power required to lift a 10-newton crate up to a 400-centimeter- high shelf in 2.0 seconds.

WHAT'S THE TRICK?

Power is the rate that work is done. Solve for work using the methods described in the previous section, and then divide by time. In this case, force and displace­ment are given. However, displacement is in units of centimeters. You must first convert centimeters into meters.

Find the Power at a Constant Velocity

Determine the power required to lift a 10-newton crate vertically at a constant speed of 2.0 meters per second.

WHAT'S THE TRICK?

This problem is similar to Example 7.8. However, this time the speed is given instead of the distance and time.

P= Fv
P = (10 N)(2.0 m/s) = 20 W

CONSERVATION OF ENERGY
Energy is always conserved in an isolated system. The amount of energy present at the start of a problem must remain constant throughout the entire problem. You must consider some important properties of energy to understand conservation of energy fully.

1. Energy can transform from one type to another.
2. Energy can transfer from one object to another.
3. Energy can leave and enter a closed system as work.

Questions concerning the conservation of energy will ask you to account for all of the joules of energy being transferred and transformed. The unit for energy, joules, can be used for problems involving kinetic energy, gravitational potential energy, elastic potential energy, electrical energy, light energy, nuclear energy, heat, and work. Typically, questions will require you to account for where the energy goes in the given scenario.

Conservative Forces
The force of gravity, F , and the force of springs, F_s, are examples of conservative forces. When conservative forces act, the total mechanical energy in a system remains constant. Take, for example, a roller coaster. At the top of a hill, the roller coaster will have the maximum gravi­tational potential energy, U_g = mgh. Should the roller coaster be moving as well, it will also possess kinetic energy, K=The sum of these energies, K+ U, is the total mechanical energy of the roller coaster. As the roller coaster descends the hill, it will lose height and gravitational potential energy. However, the roller coaster will simultaneously gain speed and kinetic energy. During the descent, energy transforms from gravitational potential energy into kinetic energy. The force of gravity is a conservative force. When conservative forces act, the total mechanical energy of a system (the roller coaster) remains constant.

K_i + U_i = K_f + U_f

Nonconservative Forces
When nonconservative forces act, energy transfers into or out of a system. As a result, the total mechanical energy of the system is not conserved. At first glance, this seems to violate the conservation of energy. When energy is gained by the system, the energy comes from the environment. When energy is lost by the system, the energy moves to the environment. Energy is conserved when the system and environment are examined together. However, most problems deal with only a specific system. When nonconservative forces act, the energy of a system is not conserved. The transfer of energy into and out of a system is known as the work of nonconservative forces. It is equal to the change in energy of the system.

W_{nonconservative force} = ΔF_sys
W_{nonconservative force} = E_f - E_i

The most commonly encountered nonconservative force is kinetic friction, f_t Kinetic friction acts on moving objects, such as the roller coaster described in the section “Conservative Forces.” When kinetic friction acts, some of the total mechanical energy of the system (roller coaster) is lost (not conserved). Kinetic friction transfers a portion of the initial energy to the environment as heat, resulting in a lower final energy for the system. As a result, the final kinetic energy and speed are less than they would have been in a frictionless environment. The decrease in kinetic energy when kinetic friction slows a moving object is referred to as kinetic energy lost. As with other forms of work, the work done by kinetic fric­tion is equal to the product of the force of kinetic friction and displacement. In addition, the work done by friction is equal to the kinetic energy lost by the system and the resulting heat transfer to the environment.

W_f = - f_kd

Conservation of Energy When Conservative Forces Act

A boy standing on a ledge throws a ball with a mass of 0.50 kilograms straight up into the air with an initial velocity of 20 meters per second. The ball is initially 5.0 meters above the ground. It rises to a maximum height and then falls, missing the ledge on the way down. The ball impacts the ground 5.0 meters below its initial starting point.
(A) What is the total mechanical energy of the ball as measured from the ground?

WHAT'S THE TRICK?

All of the energy is present in the ball at the moment the ball is released. Sum the kinetic energy of the throw and the gravitational potential energy due to the height above the ground. The ground is the lowest point reached by the bail. Set the ground as zero height and zero gravitational potential energy.

(B) What is the total mechanical energy of the bail 2.0 meters above its release point?

WHAT'S THE TRICK?

The total mechanical energy is conserved and constant.

E_i = E_{any point} = 125 J

(C) What is the maximum height reached by the ball as measured from the ground?

WHAT'S THE TRICK?

The kinetic energy of the boy’s throw will be converted into potential energy at the top of the ball’s trajectory. In addition, when an object reaches maximum height, its vertical speed is zero.
Method 1: Energy Solution

The left side of the equation was calculated in part (A), and the final velocity is zero.

125 = 0 + (0.50 kg)(10 m/s)h_f
h_f = 25 m

The height of the ledge was included in the initial energy. Therefore, the calculated final height is the height above the ground.

Method 2: Kinematics Solution
Time is unknown. Use the kinematic equation that does not include time.

This is the height as measured from the ground. The height of the ledge was included as the initial height, y_i = 5 m.
(D) What is the impact velocity when the ball reaches the ground?

WHAT'S THE TRICK?

Again, the total mechanical energy remains constant. When the ball reaches the ground, the height and gravitational potential energy will be zero, mgh_f = 0.
All of the ball’s initial energy will be completely converted into kinetic energy.

Conservation of Energy and Pendulums

A 0.02-kilogram pendulum bob is released from position A. As the pendulum passes position C, located 1.25 meters below position A, what is its velocity?

WHAT'S THE TRICK?

In position A, the pendulum is at rest and is located at the maximum height. All of the energy is stored as gravitational potential energy. In position C, the height and gravitational potential energy are zero. Potential energy has been entirely transformed into kinetic energy. The total mechanical energy remains constant.

Rearrange and simplify.

This equation is the same as the equation for an object dropped straight down from a height h. How is this possible? The pendulum bob does fall a height h, but it is attached to a string that changes the pendulum’s direction. The string changes the direction of the pendulum bob without affecting the change in energy from potential to kinetic.

Conservation of Energy and Springs

A 1.0-kilogram block moves with a speed of 10 meters per second and strikes a spring that has a spring constant of 25 newtons per meter. Determine the maximum compression of the spring.

WHAT'S THE TRICK?

Initially, the moving block has only kinetic energy and the spring is at rest. When the block strikes the spring, the spring compresses and the block slows. The kinetic energy of the moving block is transferred to the spring and stored as elastic potential energy. At maximum compression, all of the kinetic energy initially present has been entirely converted to elastic potential energy.

Substitute values and solve.

Energy Lost by Nonconservative Forces

A 50-kilogram skier, starting at rest, skis down a 5.0-meter-tall hill inclined at 30°. The skier arrives at the bottom with 2,000 joules of kinetic energy. How much energy was lost due to friction?

WHAT'S THE TRICK?

If the hill were frictionless, the skier should arrive at the bottom with the kinetic energy equal to the initial potential energy. When friction acts, energy is lost from the system and transfers to the environment as heat. Energy lost due to friction is the difference between the expected and the actual kinetic energy.

At the bottom of the hill, the skier has an actual kinetic energy of 2,000 J. The difference between the expected and the actual kinetic energies is the energy lost due to friction.

K_lost = (2,500 J) - (2,000 J) = 500 J