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# Periodic Motion

Periodic motion is a motion that repeats at certain time intervals, such as a rocking chair moving back and forth. Another example would be a swinging pendulum. The time it takes for the pendulum to complete one oscillation is called the period ‘T’, which is usually measured in seconds. The term ‘period’ is used to describe the time it takes for an event to happen, regardless of…

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# Periodic Motion

Periodic Motion
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Nie wieder prokastinieren mit unseren Lernerinnerungen. Periodic motion is a motion that repeats at certain time intervals, such as a rocking chair moving back and forth. Another example would be a swinging pendulum. The time it takes for the pendulum to complete one oscillation is called the period T’, which is usually measured in seconds. The term ‘period’ is used to describe the time it takes for an event to happen, regardless of whether that event is repetitive or not.

However, we are here concerned with repetitive events, as periodic motion is repetitive. We, therefore, also need to take into consideration the frequency ‘f . This specifies the number of times the event happens in one unit time (usually a second), and it is measured in Hertz.

All events that repeat after a certain amount of time (period) are said to be periodic, and all periodic motions have a frequency. The relationship between frequency and period is expressed in the following equation:

$f = \frac{1}{T}$

## What are the types of periodic motion?

While periodic motion can be seen in a variety of events, it is helpful to distinguish between specific types of motion such as circular motion and simple harmonic motion.

### Circular motion

Circular motion occurs around a central point with a constant radius and a certain speed. There are a variety of examples of circular motion in our daily lives, such as the blades of a ceiling fan rotating or a giant carousel wheel at the park.

While the circular motion is present, the object’s distance from the axis of rotation, also known as radius, stays constant at all times. The velocity of the object, however, changes continuously since velocity is a vector quantity that depends on both speed and direction.

The change in velocity indicates the presence of an acceleration, which is caused by a centripetal force. The centripetal force is the force that maintains the circular motion, acting towards the direction of the centre and along the radius.

$F = \frac{m \cdot v^2}{r}$

• F is the centripetal force in Newtons.
• m is the mass of the object in kg.
• v is the velocity of the object in m/s.
• r is the radius of the objects orbit. Fig. 1 - A diagram of circular motion

The rate of change of the object’s angular position is the angular velocity (ω). If an object is moving, as shown in Figure 1, in a circular motion whose radius is r, with the period of one rotation being T, we can determine its angular velocity using the formula below:

$\frac{d \theta}{d t} = \omega \space rad/s = \frac{2 \pi}{T} = 2 \pi ƒ$

Here, f is the frequency in Hertz.

The relation between the object’s speed (v) and its angular velocity can be expressed as follows:

$v = \omega \cdot T$

The angle covered in a time period (t ) is also dependent on the angular velocity (ω) and can be expressed as shown below (the values are given in radians).

$\theta = \omega \cdot t$

A tennis ball connected to a stick with a 1.2m long rope is following a circular motion around the stick at a constant velocity. The tennis ball has a mass of 60g and is moving at a velocity of 28.2 m/s. Calculate the centripetal force.

Solution:

You have learned that the equation for the centripetal force is:

$F = \frac{m \cdot v^2}{r}$

The length of the rope (r) is 1.2m, the mass of the ball (m) is 60g, and the velocity (v) is 28.2 m/s. Let’s put these variables into the equation to calculate F.

$F = \frac{(0.06 \space kg) \cdot (28.2 \space m/s)^2}{1.2 \space m} = 39.7 \space N$

### Simple harmonic motion

Simple harmonic motion is the name given to a system that can be explained using Hooke’s law. This states that an object’s displacement or deformation size is directly proportional to the load or force applied to it. Any system obeying this law is known as a simple harmonic oscillator.

Hooke’s law states that the force it takes to extend or compress a spring is directly proportional to the distance the spring is going to extend or compress and the force constant k, which is a constant factor characteristic of the spring.

$F[N] = k [N/m] \cdot x [m]$ (N: Newton, m: meter)

If there is no friction nor any other external force acting on the object, it will oscillate with equal displacement on either side of the position the oscillator takes when there are no forces acting on it. Fig. 2 - A diagram of a simple harmonic oscillator

To calculate the period or frequency of simple harmonic oscillators, we need to consider the object’s mass m and the force constant of the spring k, but we don’t need to consider the amplitude of the applied force, which is independent of the period.

For instance, regardless of whether you pluck the strings of a guitar hard or more softly, they will oscillate with the same frequency because the period is constant. The period, mass, and spring constant relationship are given by the formula below.

$T = 2 \pi \cdot \sqrt{\frac{m}{k}}$

• m is the mass of the object in kg.
• k is the force constant in N/m.
• T is the oscillation period in seconds.

You can also express this using the frequency rather than the period. If the frequency f is equal to 1/T, we obtain the following expression:

$ƒ = \frac{1}{2\pi} \cdot \sqrt{\frac{k}{m}}$

A cube with a mass of 4kg is attached to a string with a force constant of 2 N/m. Calculate the period and the frequency of the cube if a force of 10N is applied to compress the string.

Solution:

The mass of the cube m, the force constant k, and the force F have all been specified. We need to remember that the period or the frequency of simple harmonic motion is independent of the force applied.

As we said, the equation for finding the period of simple harmonic motion is:

$T = 2 \pi \cdot \sqrt{\frac{m}{k}}$

Adding the known variables, we get:

$T = 2 \pi \cdot \sqrt{\frac{4}{2}} = 2 \cdot 3.14 \cdot 1.41 = 8.85 \space s$

To find the frequency of the motion, we can either use the equation for finding the frequency or take a shortcut by remembering that the frequency f is equal to 1/T.

$ƒ = \frac{1}{T} = \frac{1}{8.85} = 0.113 \space Hz = 113 \space mHz$

### The simple pendulum

There are a variety of applications for pendulums in our daily lives. Some are very important, like the pendulums used in clocks, while others are for fun, such as a children’s swing in a playground. For small displacements, a simple pendulum can be considered a simple harmonic oscillator.

A simple pendulum is an object with a certain mass suspended from a string or wire. Fig. 3 - Diagram of a simple pendulum and the forces acting on it

As you can see in Figure 3, the linear displacement from the equilibrium is s, the gravitational force is w = mg, where m is mass and g is the acceleration due to gravity in m/s², while T is the tension force from the string that is keeping the object connected to the beam.

We can express the net force (F) acting on the pendulum as follows:

$F = -mg \cdot \sin(\theta)$

The relation between s and L is:

$s = L \cdot \sin(\theta)$

You can see from this equation that θ is equal to s/L. If we add this to the equation for finding the net force and take sin (θ) as sin (θ) ≈ θ, we can express it as follows:

$F \approx -\frac{mg}{L} \cdot s$

For small angles of oscillation (θ), the value of sin (θ) is approximately equal to θ itself.

This is another form of F = -kx, and it tells us that we can take k = mg/L when the displacement is x = s. This helps us to express the period in another way.

As we know, the period of a simple harmonic oscillator can be determined as follows:

$T = 2 \pi \cdot \sqrt{\frac{m}{k}}$

If we replace k with mg/L, we get:

$T = 2 \pi \cdot \sqrt{\frac{m}{mg/L}}$

Thus,

$T = 2 \pi \cdot \sqrt{\frac{L}{g}}$

Find the acceleration due to gravity for a simple pendulum with a length of 50 cm and a period of 1.4576 s.

Solution:

If we square the equation for finding the period and solve it for g, we can find the acceleration due to gravity.

As we know, the equation for finding the period is:

$T = 2 \pi \cdot \sqrt{\frac{L}{g}}$

If we square both sides and solve it for g, we get:

$$g = 4 \pi^2 \cdot \frac{L}{T^2}$$

\(g = 4 \pi^2 \cdot {0.5 \space m}{(1.4576 \space s)^2 = 9.281

## Periodic Motion - Key takeaways

• Periodic motion is a motion that repeats itself at certain time intervals.
• While periodic motion can be seen in a variety of events, we study periodic motion in more specific types of motion, such as circular motion and simple harmonic motion.
• The period of simple harmonic motion is independent of the amplitude of the force being applied.

No, periodic motions are not all simple harmonic motions. They can also be circular motions.

We can measure time by using periodic motion if we know the period of the motion, which is the time it takes for the motion to complete one cycle. After that, we can count the periods and multiply the counted value with the value of the period to measure time in seconds.

No, all periodic motions are not oscillatory, but all oscillatory motions are periodic motions because they repeat themselves at certain time intervals called periods.

## Periodic Motion Quiz - Teste dein Wissen

Question

What is the definition of a radian?

It is the angle subtended at a circle’s centre by an arc of equal length to the radius of a circle.

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Question

What is the name of the force that induces circular motion?

Centripetal force.

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Question

What is the direction of the velocity in circular motion compared to the circular path?

It is perpendicular to the circular path.

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Question

What is the centrifugal force?

It is a pseudo force felt by an object in circular motion.

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Question

What units do we use for the time period in circular motion?

Seconds.

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What units do we use for frequency in circular motion?

Hertz.

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Question

What is the friction force equal to in a car moving along a curve?

To the centripetal force.

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Question

What units do we use for angular speed?

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Question

What is the angular speed of an object if it takes 12 seconds to complete an orbit?

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Question

What is the angular speed of an object if it takes 3 seconds to complete three orbits?

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Question

If an object A takes less time to complete an orbit than an object B, which one has a larger angular velocity?

Object A.

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Question

Can angular displacement be larger than one orbit (2π or 360degrees)?

Yes, it can.

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Question

The centripetal force exerted on a ball of 1.5kg moving in circular motion is equal to 122 Newtons. Calculate the acceleration of the ball. Then, knowing that the ball is moving 3 metres from the centre of rotation, determine its velocity.

v = 15.62m/s.

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Question

Which of the following is the symbol for period in seconds?

T.

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Question

Which of the following is the name given to the number of times an event happens in one unit time?

Frequency.

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Question

While the circular motion is present, what happens to the object’s distance from the axis of rotation?

It stay constant at all times.

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Question

Which of the following is not one of the types of periodic motion?

Linear motion.

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Question

A tennis ball connected to a stick with a 1.5m long rope is following a circular motion around the stick at a constant velocity. The tennis ball has a mass of 70g and is moving at a velocity of 18.4 m/s. Calculate the centripetal force.

15.8 [N].

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Question

A cube with a mass of 6kg is attached to a spring with a force constant of 4 N/m. Calculate the period and the frequency of the cube if a force of 50N is applied to compress the string.

7.7 [s].

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Question

Find the acceleration due to gravity for a simple pendulum with a length of 90cm and a period of 2.5246s.

5.57 m/s².

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Question

For small displacements, a pendulum can be considered a simple harmonic oscillator. True or false?

True.

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Question

What is the symbol for the force constant?

k.

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Question

Periodic motion is a motion that repeats itself at certain time intervals. True or false?

True.

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Question

The velocity of an object in a circular motion stays constant. True or false?

False.

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Question

Which law is used to explain simple harmonic motion?

Hooke’s law.

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Question

Which of the following is not a condition for an object to perform simple harmonic motion?

A) Oscillations are periodic.

B) The frequency of the object oscillating is proportional to its velocity.

C) The acceleration of the object oscillating in a simple harmonic motion is proportional to its displacement, but has an opposite direction.

The frequency of the object oscillating is proportional to its velocity.

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Question

How is Hooke’s law related to simple harmonic motion?

When a spring oscillates it performs simple harmonic motion.

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Question

What is Hooke’s law?

The force required to extend or compress a spring from its initial resting position is proportional to a constant k and its displacement.

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What is simple harmonic motion?

It is a repetitive back and forth motion of a mass on each side of an equilibrium.

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Question

What is a phase shift?

When the initial position of a harmonic oscillator at the initial time is not equilibrium, and initial velocity is not zero, then the cosine function is shifted by an angle φ.

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Question

What is the phase shift of an oscillator that started oscillating at the maximum negative position?

3π/2

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Question

What is the phase shift of an oscillator that started oscillating at the maximum positive position?

π/2

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Question

A pendulum oscillates with an amplitude of 0.8m and a frequency of 5Hz. Find the velocity of the pendulum when it is positioned 0.75 from the initial position.

8.74 m/s

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Question

What is energy in simple harmonic motion?

It is the energy that an oscillator has when it performs simple harmonic motion.

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Question

What is the total energy in simple harmonic motion?

It is the sum of kinetic and potential energy of an oscillator.

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Question

What is the kinetic energy of an oscillator?

The energy that is acquired when an oscillator is in motion.

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What is the potential energy of an oscillator?

The energy that is stored in the oscillator when it has been displaced from the equilibrium position.

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Question

What is the average energy in SHM?

It is the total energy in SHM in one time period.

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Question

Which of the following is true for an oscillator in SHM?

The energy of an ideal oscillator is constant.

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Question

How are kinetic and potential energies represented in an energy vs time graph?

By periodic functions.

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Question

Which of the following properties of the energy vs time graph is not true?

Energy is always negative when an oscillator reaches maximum amplitude.

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Question

When is the potential energy at its maximum in SHM?

When the oscillator is at its maximum amplitude positions where x = +-Xmax.

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Question

Which of the following is true for an oscillator in SHM?

Potential energy is represented by a U-shaped curve.

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Question

An oscillator weighing 1Kg is performing SHM connected to a spring that has a constant of 70 N/m. Its position is given by the equation x(t) = 12cos(5t). Determine its potential energy at the amplitude position.

2880J.

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Question

What is a free vibration?

A free vibration involves no transfer of energy between an object and its surroundings.

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Question

What is a forced vibration?

A forced vibration is the result of a periodic external driving force acting on a system.

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Question

When does resonance occur?

Resonance occurs when the driving frequency becomes equal to the resonant frequency of the system.

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Question

What is the frequency of an object performing a free vibration called?

An object performing a free vibration oscillates at its natural frequency.

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Question

What is a damping force?

A damping force acts to reduce the amplitude of an oscillatory motion.

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Question

Give an example of a damping force.

Examples include air resistance or friction between moving parts.

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