# Reviewing Mechanics Concepts

Before starting to study E&M, you should have a solid grasp of the following concepts:

Most of the understanding needed of the mentioned topics is surface level and already given by the AP Physics C formula sheet. This page will review those topics as well as point to which E&M units require an understanding of the mentioned Mechanics topics.

# Calculus

## Derivatives

Know the power rule, chain rule, and trigonometric derivatives, as listed on the formula sheet.

## Integrals

Know the integrals of the same type of functions listed previously, also shown on the formula sheet.

# Physics

## Force Vectors and Newton's Laws

A force vectors is an arrow that originates at some point and points away towards some direction. That origin point represents the object in question, the vector represents a force acting on that object, and the direction and magnitude of the vector represent the direction and "strength" of that force. Forces can be split into x and y components (or any other perpendicular set of vectors that is convenient) to make certain calculations easier. Forces can be added to represent the combined force.

Newton's Laws of Motion tell us:

1. An object at rest remains at rest and an object in motion remains in motion unless acted upon by an unbalanced force. That is to say, net acceleration of an object is zero when the net force on that object is zero. Thus, the velocity of that object remains constant, whether that is zero or a non-zero value.

2. Force is equal to mass times acceleration. Continuing with the concept from the first law, force and acceleration can be directly computed through the F=ma equation, where a is the acceleration caused by that force.

3. For every action there is an equal and opposite reaction. This means a force imposed by object A on object B will also see a force by object B on object A. In E&M this is seen with Coulomb's Law, two current-carrying-wires in parallel, and Lenz's Law.

Present in E&M Units:

• Unit 1: Kinematics with charged particles in electric fields.

• Unit 7: Kinematics with charged particles in magnetic fields.

• Unit 8: Two current-carrying wires in parallel will exhibit magnetic forces equal but opposite in direction to each other.

• Unit 9: Newton's Third Law explanation of Lenz's Law.

## Kinematics

Kinematics describes how an object's movement and position is impacted by acceleration and velocity. The formulas on the right (which are on the formula sheet) can all be derived from the understanding that velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time. Like forces, velocity and acceleration are vectors that can be split into perpendicular components.

Present in E&M Units:

• Unit 1: Kinematics with charged particles in electric fields.

• Unit 7: Kinematics with charged particles in magnetic fields.

## Gravitational Force and Potential Energy

On the surface of the Earth, objects face a constant downward gravitational force equal to m*g, where g is around 10 meters per second-squared. We can extend this understanding to potential energy, saying that objects at a certain height h will have a gravitational potential energy of mgh.

Present in E&M Units:

• Unit 1: Some kinematics problems with charged particles in electric fields also incorporate the gravitational force.

• Unit 3: Some electrical potential energy problems also incorporate gravitational potential energy.

## Conservation of Energy and Work

As mentioned, objects can possess potential energy (UE), which is derived from the object's position with respect to other objects. In E&M, you will mostly see this with respect to a charged object's position relative to an electric field. Objects can also have kinetic energy (KE), given by the equation KE = (1/2)mv^2, where v is the velocity of that object. For physics' purposes, energy is 100% conserved, meaning that the total UE and KE of a system (in many cases, just a singular object) will remain constant throughout time, but energy is free to "change" between being kinetic and potential energy of that object. As a formula, this idea can be expressed by the equation on the right.

The change in energy of an object or system is called the Work done on that object, and it can also be expressed by the integral of a Force's component in the direction of the motion through the distance of which the force is applied. This is summarized by the following equation on the right given in the formula sheet.

Finally, power is the rate at which Work is done or that a change in energy occurs. It is given by dW/dt.

Present in E&M Units:

• Unit 3: Many problems involving electric potential energy require an understanding of conservation of energy and sometimes the work definition of power.

## Circular and Rotational Motion

When a net force is constantly perpendicular to the velocity of an object, it does not change the speed (magnitude of velocity) of the object, and thus does no work, However, it does constantly change the direction of the object, and if this force (called a centripetal force) is always to the same side of the velocity, it will cause the object to move in a circular path. This force causes a centripetal acceleration, which can be derived to be equal to a = (v^2)/r, or the velocity-squared over the radius of the circular path of motion.

If it is assumed that an object is moving in circular motion, then one can calculate its angular velocity, which is measured in radians per second instead of meters per second. Angular velocity and angle are entirely analogous to velocity and directional displacement.

Finally, period (T) is the time taken for an object in rotational motion to complete a single revolution. It can be calculated either by the circumference divided by velocity, or 2*pi divided by the angular velocity.

Present in E&M Units:

• Unit 1: Very rarely, a charged particle will be in revolutionary motion around another charged particle, such that the electric force is centripetal and thus rotational motion questions can be asked.

• Unit 7: Given that the magnetic force acts as a centripetal force, most questions will require an understanding of rotational motion.

• Unit 9: Sometimes, induced emf/current will be produced from a rotating loop of wire, whose values will be modeled by the rotational motion of the loop.

## Torque

Torque is the rotational motion analog of a linear force. For the level of understanding needed for E&M, the formula provided on the formula sheet is sufficient. The magnitude of torque is given by the magnitude of the cross product of the linear force and the radius (distance) of the force from the axis of revolution/rotation. Torques can be added to create a net torque in the same way linear forces can be added to produce a net force.

Present in E&M Units:

• Unit 7: Loops of wires in certain orientations in a magnetic field will experience a net torque.

## Oscillatory Motion

For E&M, all that needs to be understood about oscillatory motion is that it is characterized by repetitive "back-and-forth" motion. It is commonly represented by a sinusoidal function as shown.

Present in E&M Units:

• Unit 1: charged particles can move in oscillatory motion in certain electric field configurations

• Unit 10: The voltages, charges, currents, and stored energies of components in LC circuits exhibit oscillatory behavior.