PHYS101: Introduction to Mechanics

6a. Calculate the work done on an object by a force

• Define work.
• What is the unit used for work done on an object by a force?
• Use the equation describing work to calculate the work done on an object by a force.

Work is done on a system when a constant applied force causes the system to be displaced or moved in the direction of the applied force. We can describe work using the following equation:

$w = Fd\cos\theta$

where $F$ is force, $d$ is displacement, and $\theta$ (the Greek letter theta) is the angle between $F$ and $d$.

From the equation for work, we can see that the unit for work must be Newton meter, as the unit for force is Newton and the unit for displacement (distance) is meter. We define the Newton meter as the unit joule. We use joules as the unit for work and energy.

Review a worked example of calculating work for a given force and displacement in Example 7.1: Calculating the Work You Do to Push a Lawn Mower across a Large Lawn. This example also asks you to convert from joules to the commonly used energy unit kilocalories (kcal).

6b. Use the relationship between work done and the change in kinetic energy to make calculations

• Define net work and kinetic energy.
• Calculate the kinetic energy of an object given mass and velocity.
• Calculate work done on an accelerating object.

We define kinetic energy as the energy associated with motion. Kinetic energy is calculated as 

$\mathrm{KE} = \frac{1}{2}mv^2$.

When work is done on a system, energy is transferred to the system.

We define net work as the total of all work done on a system by all external forces. We can think of the of the sum of all the external forces acting on a system as a net force, or $F_\mathrm{net}$.

We can write the equation for net work in a similar way to how we wrote the equation for work in learning outcome 6a:

$w_\mathrm{net} = F_\mathrm{net}d\cos\theta$

where $w_\mathrm{net}$ is net work, $F_\mathrm{net}$ is net force, $d$ is displacement, and $\theta$ is the angle between force and displacement.

Review an example of the forces on a box going across a conveyor belt in Figure 7.4. In this figure, we see different forces acting on the box. First, gravitational force is always present, which affects the weight ($w$) of the box. The normal force ($N$) balances the weight of the box. There is the applied force of the moving conveyor belt going to the right. Lastly, there is a horizontal frictional force from the rollers on the conveyor belt going back to the left. The weight and normal force cancel out. Therefore, the net force is the applied force minus the frictional force.

Review a worked example of calculating the kinetic energy for this box on a conveyor belt in Example 7.3: Determining the Work to Accelerate a Package. 

Review a worked example in which the net force is calculated and used to determine the net work for the same system of the box on the conveyor belt in Example 7.4: Determining Speed from Work and Energy.

6c. State the work-energy theorem

•  What is the work-energy theorem?

The work-energy theorem states that the net work on a system is the change of $\frac{1}{2}mv^2$. That is: 

work $= \frac{1}{2}mv^2_f - \frac{1}{2}mv^2_i$

where $m$ is mass, $v_f$ is final velocity, and $v_i$ is initial velocity. 

Review the section Net Work and the Work-Energy Theorem.

6d. Describe the concept of potential energy and how it relates to work

• Define conservative force, potential energy, and mechanical energy.
• Calculate the potential energy of a spring.
• What is meant by conservation of mechanical energy?
• Use conservation of mechanical energy to calculate the velocity (or speed) of an object.

A conservative force is a force that only depends on the beginning point and the end point of the system. That is, it does not depend on the path the system takes to get from beginning to end.

We define potential energy as stored energy due to a system's position. An example of an object with high potential energy is a compressed or stretched spring. When you let go of the compressed or stretched spring, the spring will release its potential energy as kinetic energy and go back to its usual shape.

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

To calculate the potential energy of a spring, $\mathrm{PE_s}$, we use the following equation:

$\mathrm{PE_s} = \frac{1}{2}kx^2$

where $k$ is the spring constant and $x$ is displacement.

Review an example of a spring being stretched in Figure 7.10. The figure shows the work and potential energy associated with this.

Conservation of Mechanical Energy states that the sum of potential energy ($\mathrm{PE}$) and kinetic energy ($\mathrm{KE}$) is constant for a given system if only conservative forces act upon the system. We can write this in two different forms:

$\mathrm{KE + PE = Constant}$

$\mathrm{KE_i + PE_i = KE_f + PE_f}$

The second version of the equation can be more useful in describing changes from initial conditions ($\mathrm{KE_i}$ and $\mathrm{PE_i}$) to final conditions ($\mathrm{KE_f}$ and $\mathrm{PE_f}$). 

Review the derivation of the conservation of mechanical energy from the work-energy theorem in equations 7.43, 7.44, 7.45, 7.46, 7.47, and 7.48 in Conservative Forces and Potential Energy.

Review a worked example of using conservation of mechanical energy to determine an object's speed in Example 7.8: Using Conservation of Mechanical Energy to Calculate the Speed of a Toy Car. In this example, we use the conservation of mechanical energy and the definitions of potential and kinetic energy to determine velocity. In these types of problems, it can be helpful to make a list of the information given in the problem to help determine what variable you can solve for.

6e. Compare and contrast conservative and non-conservative forces

• What are non-conservative forces? How does this differ from conservavive forces? 
• Give examples of conservative forces and non-conservative forces.

A non-conservative force is a force that does depend on the path taken of the object. That is, a non-conservative force depends on how an object got from its initial state to its final state. Non-conservative forces change the amount of mechanical energy in a system. This differs from conservative forces, which do not depend on the path taken from initial to final state and do not change the amount of mechanical energy in a system. 

An important example of a non-conservative force is friction. Friction is the force between two surfaces. We see friction when rolling a ball on a carpet versus a hardwood floor. The ball rolls farther on the hardwood floor than it does on a carpet. This is because the fuzzy carpet has more friction than the smooth hardwood. Friction converts some of the kinetic energy of the ball to thermal energy, or heat. As kinetic energy is converted to thermal energy, the balls slows to a stop. 

Conservative forces exist in ideal systems with no friction. An idealized spring that does not experience friction would be an example of conservative forces.

Review Figure 7.15 for a comparison of conservative and non-conservative forces. In Figure 7.15 (a) a rock is being "bounced" on an ideal spring with no friction. The mechanical energy does not change and the rock will continue bouncing indefinitely. In Figure 7.15 (b) the rock is thrown and lands on the ground. When it hits the ground, its kinetic energy is converted to thermal energy and sound. The rock can not "bounce" back up because its mechanical energy is not conserved.

6f. Solve dynamics problems using conservation of energy

• State the Law of Conservation of Energy.
• What are some examples of other energy?
• List the problem solving steps for solving dynamics problems.

The Law of Conservation of Energy states that the total energy in any process is constant. Energy can be transformed between different forms, and energy can be transferred between objects. However, energy cannot be created or destroyed. This is a broader law than the conservation of mechanical energy because this applies to all energy, not just energy when only conservative forces are applied.

We can write the Law of Conservation of Energy in the following ways:

$\mathrm{KE + PE + OE = Constant}$

$\mathrm{KE_i + PE_i + OE_i = KE_f + PE_f + OE_f}$

In the second equation, the $\mathrm{KE_i}$, $\mathrm{PE_i}$, and $\mathrm{OE_i}$ are initial conditions and $\mathrm{KE_f}$, $\mathrm{PE_f}$, and $\mathrm{OE_f}$ are final conditions. The new term, $\mathrm{OE}$, is other energy. This is a collected term for all forms of energy that are not kinetic energy or potential energy.

Other energy includes forms such as thermal energy (heat), nuclear energy (used in nuclear power plants), electrical energy (used to power electronics), radiant energy (light), and chemical energy (energy from chemical reactions). 

When solving Conservation of Energy problems, it is important to identify the system of interest, and all forms of energy that can occur in the system. To do this, we need to first identify all forces acting on the system. Then, we can plug equations for different types of energy into the Law of Conservation of Energy equation to solve for the unknown in the problem.

Review Problem Solving Strategies for Energy for a step-by-step guide for solving these types of problems.

6g. Illustrate what is meant by power

• Define power. What is the unit for power?
• Calculate power for a given system given energy and time.

We define power as the rate at which work is done. We can write this as:

$P = \frac{w}{t}$

where $w$ is work and $t$ is time.

The unit for power is the watt, W, with one W = one joule/second.

Higher power means more work is done in a shorter time. This also means that more energy is given off in a shorter time. For example, a 60 W light bulb uses 60 J of work in a second, and also gives off 60 J of radiant and heat energy every second. 

See a worked example of calculating the power of an object in motion in Example 7.11: Calculating the Power to Climb Stairs. In this problem, the power of a person going up a flight of stairs is calculated. First, the work of going up the stairs is calculated using the equation: work $\mathrm{= KE + PE}$. Then, power is calculated given the time it took the person to go up the stairs.

Review this material in Power and Work, Energy, and Power in Humans.

Unit 6 Vocabulary

• Chemical energy
• Conservation of mechanical energy
• Conservative force
• Electrical energy
• Fossil fuels
• Friction
• Joule
• Kinetic energy
• Law of Conservation of Energy
• Mechanical energy
• Net force
• Net work
• Newton meter
• Nonconservative force
• Nonrenewable energy
• Nuclear energy
• Other energy
• Potential energy
• Power
• Radiant energy
• Renewable energy
• Spring
• Thermal energy 
• Watt
• Work
• Work-energy theorem