Simple Harmonic Motion (DP IB Physics)

Simple harmonic motion (SHM) is a type of oscillation where the restoring force is directly proportional to the displacement and directed towards the equilibrium position.

True.

In SHM, the time interval of each complete vibration, known as the period, is constant.

The two properties of the restoring force in simple harmonic motion are that:

• it is always directed towards the equilibrium position

• it is proportional to the displacement from the equilibrium position

Examples of objects undergoing SHM are:

• The pendulum of a clock

• A child on a swing

• A rope bridge

• A marble in a bowl

• A bowl set vibrating

• A mass oscillating on a spring

• A bungee jumper coming to a stop at the end of their fall

In SHM, acceleration is proportional to displacement and in the opposite direction to it.

The defining equation of SHM is:

Where:

• = acceleration, measured in metres per second squared (m s²)

• = angular frequency, measured in radians per second (rad s^-1)

• = displacement, measured in metres (m)

False.

The period of oscillation in SHM is independent of the amplitude for small angles of oscillation.

Angular frequency in SHM is the change of angular displacement with respect to time, measured in radians per second (rad s⁻¹).

The quantity on the vertical axis of the graph is displacement.

The quantity on the vertical axis of the graph is acceleration.

The direction of the restoring force on the mass shown in the diagram when released from its lowest vertical displacement is upwards.

The equation for the restoring force in a mass-spring system is:

Where:

• = restoring force, measured in newtons (N)

• = spring constant, measured in newtons per kilogram (N kg^-1)

• = spring extension, measured in metres (m)

The equation for the time period of a mass-spring system oscillating with simple harmonic motion is:

Where:

• = time period, measured in seconds (s)

• = mass of spring, measured in kilograms (kg)

• = spring constant, measured in newtons per kilogram (N kg^-1)

In the mass-spring time period equation, represents the spring constant, measured in N m⁻¹.

False.

The time period of a mass-spring system is independent of the gravitational field strength.

The higher the spring constant, , the stiffer the spring and the shorter the time period of the oscillation.

The restoring force is: -4.5 N.

The frequency of the mass-spring system is 0.11 Hz

A simple pendulum is a system consisting of a string and a bob that swings from side to side.

The bob is a mass at the end of the pendulum string, generally spherical and considered a point mass.

In the period of a pendulum equation, represents the length of the string from the pivot to the centre of mass of the bob.

The equation for the time period of a simple pendulum is:

Where:

• = time period, measured in seconds (s)

• = length of string, measured in metres (m)

• = gravitational field strength, measured in newtons per kilogram (N kg^-1)

True.

The time period of a simple pendulum does depend on gravitational field strength.

Small angle approximation is when the angle of oscillation is less than 10°, allowing to be approximated as .

The restoring force in a simple pendulum is the component of the weight force acting along the arc of the circle towards the equilibrium position. It is given by the equation:

The equation for angular frequency and time period is

Where:

• = angular frequency, measured in radians per second (rad s^–1)

• = time period, measured in seconds (s)

The angular frequency is 3.5 rad s^-1

The total energy of a simple harmonic motion system always remains constant and is equal to the sum of the kinetic and potential energy.

The equation for elastic potential energy is:

Where:

• = elastic potential energy, measured in joules (J)

• = spring constant, measured in newtons per metre (N m^-1)

• = extension, measured in metres (m)

True.

In simple harmonic motion, the total energy in the system remains constant and is conserved.

Kinetic energy, in the context of SHM, is the energy an object possesses due to its motion.

The relationship between kinetic energy and potential energy in SHM is:

Where:

• = the total energy, measured in joules (J)

• = potential energy, measured in joules (J)

• = kinetic energy, measured in joules (J)

At the equilibrium position, the kinetic energy of a mass-spring system is at its maximum.

As a pendulum swings down from its highest point, gravitational potential energy decreases while kinetic energy increases.

The shape of the potential energy graph of an object undergoing half an oscillation from its amplitude position is a U shape.

When a pendulum bob is at its highest point in its swing, the gravitational potential energy is at its maximum.

The quantity represented by the horizontal line on the graph is total energy.

The two starting positions the SHM equations can be taken from are:

• the equilibrium position

• the amplitude position

The equation for the displacement of an oscillator starting from the equilibrium position is:

Where:

• = displacement of the oscillator, measured in metres (m)

• = amplitude of oscillator, measured in metres (m)

• = angular frequency of oscillator, measured in radians per second (rad s^-1)

• = time of oscillation, measured in seconds (s)

True.

The maximum speed of an oscillator in SHM occurs at the equilibrium position.

The equation for the velocity of an oscillator starting from the amplitude position is:

Where:

• = velocity of oscillator, measured in metres per second (m s^-1)

• = angular velocity of oscillator, measured in radians per second (rad s^-1)

• = angular displacement of oscillator, measured in metres (m)

• = time, measured in seconds (s)

The velocity-displacement relation is defined as:

Where:

• = velocity of oscillator, measured in metres per second (m s^-1)

• = angular velocity of oscillator, measured in radians per second (rad s^-1)

• = amplitude of oscillator, measured in metres (m)

• = displacement of oscillator, measured in metres (m)

The equation for the total energy of an object in SHM at the amplitude of oscillation is:

Where:

• = total energy of oscillator, measured in joules (J)

• = mass of oscillator, measured in kilograms (kg)

• = angular frequency of oscillator, measured in radians per second (rad s^-1)

• = amplitude of oscillator, measured in metres (m)

The displacement of an oscillator starting from its amplitude is described by the cosine function:

Where:

• = displacement of oscillator, measured in metres (m)

• = amplitude of oscillator, measured in metres (m)

• = angular frequency of oscillator, measured in radians per second (rad s^-1)

• = time of oscillation, measured in seconds (s)

The phase difference, , represents the initial angle in radians at .

The first displacement-time graph represents the displacement of a SHM oscillator starting at the equilibrium position.

The displacement-time graph that represents the oscillation of a vertical mass-spring system starting from maximum displacement is:

The equation for potential energy in SHM is

Where:

• = mass of object, measured in kilograms (kg)

• = angular frequency of oscillations, measured in radians per second (rad s^-1)

• = displacement, measured in metres (m)

The equation for total energy in SHM is

Where:

• = mass of object, measured in kilograms (kg)

• = angular frequency of oscillations, measured in radians per second (rad s^-1)

• = amplitude of oscillations, measured in metres (m)

The equation for the kinetic energy-displacement relation is

Where:

• = mass of object, measured in kilograms (kg)

• = angular frequency of oscillations, measured in radians per second (rad s^-1)

• = amplitude of oscillations, measured in metres (m)

• = displacement, measured in metres (m)

True.

The total energy of an SHM system remains constant.

The equation for maximum kinetic energy is

Where:

• = mass of object, measured in kilograms (kg)

• = angular frequency of oscillations, measured in radians per second (rad s^-1)

• = amplitude of oscillations, measured in metres (m)

Potential energy is related to displacement by the equation

Where:

• = mass of object, measured in kilograms (kg)

• = angular frequency of oscillations, measured in radians per second (rad s^-1)

• = displacement, measured in metres (m)

The trigonometric identity is

Where:

• = angular frequency, measured in radians per second (rad s^-1)

• = time of oscillations, measured in seconds (s)

A phase angle in SHM is the difference in angular displacement compared to an oscillator which starts at the equilibrium position.

The equation for displacement in SHM with phase angle is

Where:

• = angular frequency of oscillations, measured in radians per second (rad s^-1)

• = amplitude of oscillations, measured in metres (m)

• = displacement, measured in metres (m)

• = time of oscillation, measured in seconds (s)

• = phase angle, measured in radians (rad)

The sine and cosine functions of displacement are out of phase by radians.

Two points on a wave are in phase when they are at the same point in their wave cycle at the same time.

The equation for velocity in SHM with phase angle is

Where:

• = angular frequency of oscillations, measured in radians per second (rad s^-1)

• = amplitude of oscillations, measured in metres (m)

• = velocity, measured in metres per second (ms^-1)

• = time of oscillation, measured in seconds (s)

• = phase angle, measured in radians (rad)

True.

A sine wave can be described as a cosine wave that lags by .

The equation for acceleration in SHM with phase angle is

Where:

• = angular frequency of oscillations, measured in radians per second (rad s^-1)

• = amplitude of oscillations, measured in metres (m)

• = acceleration, measured in metres per second squared (ms^-2)

• = time of oscillation, measured in seconds (s)

• = phase angle, measured in radians (rad)

A SHM oscillator starting from a phase angle, , will be out of phase by an angle, .

A phase angle of corresponds to a shift to the right (positive direction).