Topic 13: Oscillations

What does it mean if an object is experiencing simple harmonic motion?

. An object experiencing a restoring force, which acts towards a centre of equilibrium

. This force is proportional to the distance from equilibrium position

. Explained using the equation F = -k x

In what situations will simple harmonic motion occur?

. SHM occurs when the force is directly proportional to the displacement from the equilibrium position

. or, when the acceleration is proportional to the displacement from the equilibrium position

Equation for the acceleration of an object experiencing SHM

. a = − ω^2 x

What is angular velocity (ω)?

. The angle an object moves per unit time

. Calculated using ω = 2πf

How would you derive T (period) from angular velocity (ω)?

. T = 1/f = 2π/ω

What are the three equations for x (displacement), v (velocity) and a (acceleration) in a simple harmonic oscillator?

. x = A cos ω t

. v = -A ω sin ω t

. a = -A ω^2 cos ω t

Calculating time period in a simple pendulum:

. T = 2π √l/g

Calculating time period in a mass-spring system:

. T = 2π √m/k

Displacement-time graph for an oscillating system

. x = A cos ω t

. Will follow a sine/cosine curve with a maximum A and minimum A

. As velocity is change in displacement over time, you can find the velocity of a system at a point by taking the gradient

Velocity-time graph for an oscillating system

. v = -A ω sin ω t

What is resonance?

. When the amplitude of oscillations of a system drastically increases due to gaining energy from the driving force

. Occurs when the driving frequency (the frequency of the force driving the system) is equal to the natural frequency of the system

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What are three applications of resonance?

. Instruments: flutes have a long tube in which air resonates, causing a stationary sound wave to be formed

. Radio: tuned so their electric current resonates at the same frequency as the desired broadcast frequency

. A swing: the pushing acts as the ‘driving frequency’, which causes resonance if it is equal to the resonant frequency

What is core practical 16?

. Determine the value of an unknown mass using the resonant frequencies of the oscillation of known masses.

What is damping?

. Where a force acts on an oscillating system and energy is lost from the system to its environment, leading to reduced amplitude of oscillations

. An oscillating system cannot gain or lose energy unless an external force acts upon it: this is the principle of conservation of energy

What is a usage of damping?

. Resonance can have negative consequences, for example causing damage to a bridge when people walking across are providing a driving frequency close to the natural frequency

. This will cause amplified oscillations, which could cause damage to the bridge

. Damping decreases the effects of resonance

How does total energy remain constant in a simple harmonic oscillating system?

. Kinetic energy is transferred to potential energy and back as the system oscillates

. At max. amplitude: max. amount of potential energy

. As it moves towards equilibrium, potential energy is converted to kinetic energy so that at the centre of oscillation, kinetic energy is at a maximum

. Kinetic energy transferred to potential as object moves away from equilibrium

. In a damped system, an external force causes this total energy to decrease; for example, air resistance in a pendulum system

What is a free oscillation?

. Occur when no external force is continuously acting on the system, therefore the system will oscillate at its natural frequency

What is a forced oscillation?

. When a system experiences an external driving force which causes it to oscillate

. The frequency of the driving force is significant

What is the effect of damping on resonance?

. As the degree of damping increases, the resonant frequency decreases and the maximum amplitude decreases

What are the three types of damping?

. Light damping: where the amplitude gradually decreases by a small amount each oscillation

. Critical damping: Reduces the amplitude to 0 in the shortest possible time

. Heavy damping: Amplitude reduces slower than with critical damping, but also without any oscillations

How does the plastic deformation of ductile materials reduce the amplitude of oscillation?

. A ductile material is one that can undergo a large amount of plastic deformation before fracturing, meaning it will be permanently deformed

. The plastic deformation of a ductile material can be used to reduce the amplitude of oscillations, because energy is used to deform the material

An example of plastic deformation being used to reduce amplitude of oscillation

. A climbing rope is manufactured so that it will reduce the amplitude of oscillations as quickly as possible (through critical damping), meaning that they can stay safe and not have to bounce many times before stopping

. As a climbing rope suffers plastic deformation when a climber falls, it is permanently deformed and cannot use again

What happens when damping is decreased in a mass-spring system?

. Maximum amplitude increases

. Frequency at which maximum amplitude occurs increases

Explain how the maximum kinetic energy of an oscillating trolley will change if the amplitude of oscillation is doubled

. v = ω A since ω is constant, if A doubles, v doubles

. due to Ek = 1/2 m v^2 if v doubles, Ek quadruples

How could you modify a method to obtain time period of a simple pendulum to improve accuracy?

. Time more oscillations and divide by no. of oscillations timed to improve percentage uncertainty

. Use a fiducial marker at equilibrium so that it’s easier to determine when the pendulum passes

. Let the pendulum swing before starting the stopwatch, as the first oscillation could be affected by someone pushing the pendulum

What is the procedure to make an accurate determination of time period of an oscillating trolley?

. Time n oscillations and divide by n to find T

. Make n a large number to reduce percentage uncertainty

. Repeat and calculate a mean

. Use a fiducial marker to indicate reference position

. Make this reference position at equilibrium

. As the trolley is travelling fastest at equilibrium, uncertainty in starting and stopping the stopwatch is least here