Unit 4: Energy

# Work and Mechanical Energy

## Work

When you lift a dumbbell from the floor, you exert a force on it over a distance, and when you push a box across a floor, you also exert a force on it over a distance. The application of force over a distance is called work. Work is a scalar quantity and is measured in units of J (joules).

Work is the transfer of energy that occurs when a force is applied over a distance.

The formula for work is W = Fd, where W is work, F is force, and d is distance.

Work is measured in joules (J).

### Work at an Angle

The previous formula only works when work is done completely parallel to the intended distance of travel. When the force is done at an angle, the formula becomes:

`W = Fd cos θ`

**Example:** A force is applied to a block at an angle of 30 degrees to the horizontal. The force has a magnitude of 50 N and the block is displaced by 2 meters in the direction of the force. Using the equation that relates work, force, displacement and the angle between the force and displacement, we can find the work done on the block, which is equal to the force times the displacement times the cosine of the angle between them. Thus, the work done on the block is 86.6 J.

## Mechanical Energy

Mechanical energy is the sum of kinetic energy and potential energy in a system.

Kinetic energy is the energy of motion and is given by the formula KE = 1/2mv^2, where m is mass and v is velocity.

Potential energy is the energy stored in an object due to its position or configuration and is given by the formula PE = mgh, where m is mass, g is acceleration due to gravity, and h is height.

Mechanical energy is conserved in a closed system, meaning that the total amount of mechanical energy remains constant.

## Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.

The formula for the work-energy theorem is Wnet = ΔKE, where Wnet is the net work done on an object and ΔKE is the change in its kinetic energy.

The work-energy theorem can be used to calculate the work done on an object or the change in its kinetic energy.

`Wtotal = deltaK`

The work-energy theorem begins to answer the question by stating that a system gains or loses Kinetic Energy by transferring it to through work between the environments.

## Conservation of Mechanical Energy

In a closed system, the total amount of mechanical energy is conserved.

This means that the sum of kinetic energy and potential energy remains constant.

The conservation of mechanical energy can be used to solve problems involving the motion of objects in a system.

The sum of an object’s kinetic energy and potential energies is called its **total mechanical energy**

`E = K + U`

`Ki + Ui = Kf + Uf`

This is the simplest form of the **Law of Conservation on Total Energy.**

## Conservation of Energy with Nonconservative Forces

The equation `Ki + Ui = Kf + Uf`

holds if no nonconservative forces are doing work. However, if work is done by such forces during the process under investigation, then the equation needs to be modified to account for this work as follows:

`Ki + Ui + Wother= Kf + Uf`

**Example Questions:**

Suppose a block of mass 2 kg is placed on a rough surface with an initial velocity of 5 m/s. The coefficient of kinetic friction between the block and the surface is 0.2. The block comes to rest after covering a distance of 10 m. Find the work done by frictional force.

# Power

Power is the rate at which work is done or energy is transferred. It is a scalar quantity and is measured in watts (W). Power is the rate at which energy is transferred into, or out of, within a system,.

## Formula

The formula for power is:

`P = W/t`

where P is power, W is work, and t is time.

## Units

The SI unit for power is watts (W), which it was originally (Joules/s) later renamed the watt. Other common units include horsepower (hp) and kilowatts (kW).

## Calculations

To calculate power, you need to know the amount of work done and the time it took to do it. For example, if a person lifts a 50 kg weight 2 meters in 5 seconds, the work done is:

`W = mgh`

W = (50 kg)(9.8 m/s^2)(2 m)

W = 980 J

The power can then be calculated using the formula:

`P = W/t`

P = 980 J / 5 s

P = 196 W

Therefore, the power output of the person lifting the weight is 196 watts.

## Power and Energy

Power and energy are related, but they are not the same thing. Energy is the ability to do work, while power is the rate at which work is done. The amount of energy used depends on both the power and the time it is used for. For example, a 100 W light bulb left on for 10 hours uses more energy than a 50 W light bulb left on for the same amount of time.

## Power and Efficiency

Efficiency is a measure of how much of the input energy is converted into useful output energy. The efficiency of a device can be calculated using the formula:

`efficiency = useful output energy / input energy`

Power is also related to efficiency. The higher the power output of a device, the more energy it can convert into useful work. However, a device with a high power output may not necessarily be more efficient than a device with a lower power output.

# Unit 4: Energy

# Work and Mechanical Energy

## Work

When you lift a dumbbell from the floor, you exert a force on it over a distance, and when you push a box across a floor, you also exert a force on it over a distance. The application of force over a distance is called work. Work is a scalar quantity and is measured in units of J (joules).

Work is the transfer of energy that occurs when a force is applied over a distance.

The formula for work is W = Fd, where W is work, F is force, and d is distance.

Work is measured in joules (J).

### Work at an Angle

The previous formula only works when work is done completely parallel to the intended distance of travel. When the force is done at an angle, the formula becomes:

`W = Fd cos θ`

**Example:** A force is applied to a block at an angle of 30 degrees to the horizontal. The force has a magnitude of 50 N and the block is displaced by 2 meters in the direction of the force. Using the equation that relates work, force, displacement and the angle between the force and displacement, we can find the work done on the block, which is equal to the force times the displacement times the cosine of the angle between them. Thus, the work done on the block is 86.6 J.

## Mechanical Energy

Mechanical energy is the sum of kinetic energy and potential energy in a system.

Kinetic energy is the energy of motion and is given by the formula KE = 1/2mv^2, where m is mass and v is velocity.

Potential energy is the energy stored in an object due to its position or configuration and is given by the formula PE = mgh, where m is mass, g is acceleration due to gravity, and h is height.

Mechanical energy is conserved in a closed system, meaning that the total amount of mechanical energy remains constant.

## Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.

The formula for the work-energy theorem is Wnet = ΔKE, where Wnet is the net work done on an object and ΔKE is the change in its kinetic energy.

The work-energy theorem can be used to calculate the work done on an object or the change in its kinetic energy.

`Wtotal = deltaK`

The work-energy theorem begins to answer the question by stating that a system gains or loses Kinetic Energy by transferring it to through work between the environments.

## Conservation of Mechanical Energy

In a closed system, the total amount of mechanical energy is conserved.

This means that the sum of kinetic energy and potential energy remains constant.

The conservation of mechanical energy can be used to solve problems involving the motion of objects in a system.

The sum of an object’s kinetic energy and potential energies is called its **total mechanical energy**

`E = K + U`

`Ki + Ui = Kf + Uf`

This is the simplest form of the **Law of Conservation on Total Energy.**

## Conservation of Energy with Nonconservative Forces

The equation `Ki + Ui = Kf + Uf`

holds if no nonconservative forces are doing work. However, if work is done by such forces during the process under investigation, then the equation needs to be modified to account for this work as follows:

`Ki + Ui + Wother= Kf + Uf`

**Example Questions:**

Suppose a block of mass 2 kg is placed on a rough surface with an initial velocity of 5 m/s. The coefficient of kinetic friction between the block and the surface is 0.2. The block comes to rest after covering a distance of 10 m. Find the work done by frictional force.

# Power

Power is the rate at which work is done or energy is transferred. It is a scalar quantity and is measured in watts (W). Power is the rate at which energy is transferred into, or out of, within a system,.

## Formula

The formula for power is:

`P = W/t`

where P is power, W is work, and t is time.

## Units

The SI unit for power is watts (W), which it was originally (Joules/s) later renamed the watt. Other common units include horsepower (hp) and kilowatts (kW).

## Calculations

To calculate power, you need to know the amount of work done and the time it took to do it. For example, if a person lifts a 50 kg weight 2 meters in 5 seconds, the work done is:

`W = mgh`

W = (50 kg)(9.8 m/s^2)(2 m)

W = 980 J

The power can then be calculated using the formula:

`P = W/t`

P = 980 J / 5 s

P = 196 W

Therefore, the power output of the person lifting the weight is 196 watts.

## Power and Energy

Power and energy are related, but they are not the same thing. Energy is the ability to do work, while power is the rate at which work is done. The amount of energy used depends on both the power and the time it is used for. For example, a 100 W light bulb left on for 10 hours uses more energy than a 50 W light bulb left on for the same amount of time.

## Power and Efficiency

Efficiency is a measure of how much of the input energy is converted into useful output energy. The efficiency of a device can be calculated using the formula:

`efficiency = useful output energy / input energy`

Power is also related to efficiency. The higher the power output of a device, the more energy it can convert into useful work. However, a device with a high power output may not necessarily be more efficient than a device with a lower power output.