Forces, Acceleration, and Energy (HSC SSCE Physics): Revision Notes

Energy

Energy is one of the most important concepts in physics. It connects to our understanding of forces and helps us describe how objects move and interact. Understanding energy allows us to predict and explain a wide range of physical phenomena.

Types of energy

There are two fundamental forms of energy in physics:

Kinetic energy is the energy an object possesses due to its motion. Any moving object has kinetic energy.

Potential energy is stored energy that exists because of the forces acting between objects in a system. It depends on the positions of objects relative to each other.

Kinetic energy

Kinetic energy exists whenever an object is moving. The faster an object moves, or the more massive it is, the greater its kinetic energy.

The kinetic energy formula

For a single object moving at a particular speed, the kinetic energy is given by:

$E_k = \frac{1}{2}mv^2$

where:

• $E_k$ is the kinetic energy (measured in joules, J)
• $m$ is the mass of the object (in kilograms, kg)
• $v$ is the velocity of the object (in metres per second, $\text{m} \cdot \text{s}^{-1}$)

Units of energy

Looking at the kinetic energy formula, we can work out that the units must be $\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}$. This combination of units is given the special name joule (symbol J), in honour of physicist James Prescott Joule.

$1\text{ J} = 1\text{ kg} \cdot \text{m}^2 \cdot \text{s}^{-2}$

Important properties of kinetic energy

Kinetic energy is always positive. It doesn't matter which direction an object is moving; as long as it's moving, it has kinetic energy.

Different types of kinetic energy

In this course, we focus mainly on the kinetic energy of single, large-scale objects (like cars, balls, or books). However, kinetic energy appears in other contexts too:

• Macroscopic kinetic energy: A single object moving in one direction, like a moving car
• Wave motion: When a wave passes through a material, individual particles oscillate up and down or back and forth, giving them kinetic energy, even though there's no net movement of material
• Thermal energy: All particles in matter are constantly moving randomly. This random kinetic energy of many microscopic particles is what we measure as temperature

Potential energy

Potential energy is stored energy that has the capacity to do work. Whenever a force acts on an object, there is potential energy associated with that force.

Understanding potential energy

We can think of potential energy as being stored in the field that produces the force.

Gravitational potential energy

When you hold an object at a height above Earth's surface, you're working against the gravitational force. The Earth-object system stores gravitational potential energy. If you release the object, this stored energy gets converted into kinetic energy as the object falls and accelerates.

The gravitational potential energy of an object near Earth's surface is given by:

$U_g = F_{\text{gravitational}}h = mgh$

where:

• $U_g$ is the gravitational potential energy (in joules, J)
• $m$ is the mass of the object (in kilograms, kg)
• $g$ is the gravitational field strength, equal to the acceleration due to gravity ($9.8 \text{ m} \cdot \text{s}^{-2}$ near Earth's surface)
• $h$ is the height of the object above a chosen reference point (in metres, m)

Positive and negative potential energy

Unlike kinetic energy, potential energy can be positive or negative. We define the zero point of potential energy by choosing a convenient reference position. Once we've chosen this reference point, potential energy is calculated based on the object's distance from that position.

Other types of potential energy

Beyond gravitational potential energy, forces in other fields also store energy:

• Electric potential energy: A charged particle in an electric field experiences a force, so the field stores potential energy. If the particle is free to move, it will accelerate, converting potential energy to kinetic energy.
• Magnetic potential energy: Magnetic fields also exert forces and store potential energy.
• Elastic potential energy: A compressed or stretched spring has stored energy. When released, the spring can do work by applying a force over a distance. This is actually a form of electromagnetic potential energy, because the force from a spring comes from atoms being pushed closer together or pulled further apart than their normal separation, and atoms interact through electromagnetic forces between their electrons.

Conservation of energy

Energy is a conserved quantity. This means energy cannot be created or destroyed - it can only change from one form to another. The total amount of energy in the universe remains constant.

Isolated systems

When solving problems using conservation of energy, we need to carefully define the system we're analyzing.

An isolated system is one where energy (or mass) cannot be transferred into or out of the system. For an isolated system, the total energy stays constant:

$E_{\text{total}} = \Sigma U + \Sigma E_k = \text{constant}$

This means the sum of all potential energies plus the sum of all kinetic energies in the system doesn't change.

Non-isolated systems

Work done by a constant force

Forces act to transfer energy and to change the form of energy. When a force acts on an object and causes it to accelerate, kinetic energy increases, which means there must be a corresponding decrease in potential energy (or energy transferred from elsewhere). When a force slows an object down, kinetic energy decreases and potential energy or some other form of energy increases.

Definition of work

The amount of energy transferred by a force is called the work done by that force. Work is calculated using:

$W = F_{\parallel}s = Fs\cos\theta$

where:

• $W$ is the work done (in joules, J)
• $F_{\parallel}$ is the component of the force in the direction of motion (in newtons, N)
• $s$ is the displacement of the object (in metres, m)
• $\theta$ is the angle between the force and the displacement

Direction matters

Units of work

Since work is energy transferred, it has the same units as energy: joules (J). The formula for work also tells us that:

$1\text{ J} = 1\text{ N} \cdot \text{m}$

This makes sense because $1\text{ N} = 1\text{ kg} \cdot \text{m} \cdot \text{s}^{-2}$, so:

$1\text{ N} \cdot \text{m} = 1\text{ kg} \cdot \text{m} \cdot \text{s}^{-2} \times 1\text{ m} = 1\text{ kg} \cdot \text{m}^2 \cdot \text{s}^{-2} = 1\text{ J}$

Work from graphs

The work done by a force can be found from a graph of force versus displacement. The work done equals the area under the curve.

For a constant force, this is simply a rectangular area. When the force varies, you may need to divide the area into many small segments and add them all up.

The work-energy theorem

When work is done on an object, the object's kinetic energy changes. Specifically:

• When positive work is done (force in direction of motion), kinetic energy increases and the object speeds up
• When negative work is done (force opposes motion), kinetic energy decreases and the object slows down

The total work done on an object equals the sum of work done by all forces acting:

$W_{\text{net}} = \Sigma(F_{\parallel}s) = F_{\text{net}}s\cos\theta$

Using Newton's second law ($F_{\text{net}} = ma$) and the kinematic equation ($v^2 = u^2 + 2as$), we can show that:

$W_{\text{net}} = \frac{1}{2}mv^2 - \frac{1}{2}mu^2 = \Delta E_k$

Energy and the gravitational field

The gravitational force that Earth exerts on an object is given by $\vec{F}_{\text{gravitational}} = m\vec{g}$, where $m$ is the object's mass and $\vec{g}$ is the gravitational field strength (equal to the acceleration due to gravity).

Work done by gravity

When you drop an object, it falls and accelerates due to the gravitational force. The kinetic energy increases because gravity is doing work on the object. If we assume air resistance is negligible, then only the gravitational force acts, and we can treat the Earth-object system as isolated.

The work done by gravity when an object falls through a height $\Delta h$ is:

$W_{\text{gravity}} = F_{\text{gravitational}}s = mg\Delta h$

Looking at the expression for gravitational potential energy ($U_g = mgh$), we can see that the magnitude of work done by gravity equals the change in gravitational potential energy:

$W_{\text{gravity}} = mg\Delta h = -\Delta U_g$

Connecting work, kinetic energy, and potential energy

The work done by gravity also equals the change in kinetic energy:

$W_{\text{gravity}} = \Delta E_k = -\Delta U_g$

This can be rearranged to give:

$\Delta E_k + \Delta U_g = 0$

For an object falling freely in Earth's gravitational field:

$mg\Delta h = \frac{1}{2}m(v^2 - u^2)$

Lifting objects against gravity

When you lift an object, you exert an upward force while gravity exerts a downward force. If the object moves upward:

• You do positive work on it
• Gravity does negative work on it

Investigation: Energy changes of falling objects

Aim: To investigate the transformation of gravitational potential energy to kinetic energy as an object is accelerated by a gravitational field.

Materials:

• Objects to drop
• Data-logger equipment for measuring motion
• Computer with data-logging software

Method:

1. Set up the data-logger according to the manufacturer's instructions
2. Ensure you can record position, velocity, and acceleration data
3. Drop objects from various heights
4. Record the motion data
5. Save each dataset with a descriptive name

Analysis:

• Create graphs of velocity versus time to calculate kinetic energy as a function of time
• Create graphs of position versus time to calculate gravitational potential energy as a function of time
• Create graphs of acceleration versus time to calculate net force as a function of time
• Compare how potential energy decreases while kinetic energy increases

Discussion questions:

1. Did your graphs have the expected shapes?
2. How much gravitational potential energy was converted to kinetic energy?
3. Was air resistance significant? Did the size, shape, or mass of objects make a difference?

Conclusion: Write a conclusion addressing the aim and summarizing your findings about energy transformation.

Mechanical energy

Mechanical energy is a term used to group together:

• The kinetic energy of macroscopic objects
• Gravitational potential energy
• Sometimes elastic potential energy (energy stored in springs)

Conservation of mechanical energy

In the absence of friction and other resistive forces (like air resistance), mechanical energy is conserved. This means:

$E_{\text{mechanical}} = E_k + U_g = \text{constant}$

Reality check

When kinetic friction slows down an object, the lost kinetic energy is converted into random kinetic energy of atoms on the surfaces - the surfaces get warmer. This random microscopic kinetic energy (thermal energy) is not considered mechanical energy.

For accurate calculations, especially when friction is significant, you need to account for these energy losses.