SIMPLE HARMONIC MOTION- learn resonance, damped oscillation, forced oscillation, period, frequency angular velocity

 This is a topic that has a lot of formulae in it, as a result, we are going to take them step by step. Lets get straight into it.

First, we need to know about periodic motion. Periodic motion is the motion of a body that repeats itself at an equal interval of time. Examples are the oscillation of a clock pendulum, the vibration of the balance wheel of a watch etc.

Simple harmonic motion(SHM) is the to and fro movement of a body along a straight line such that its acceleration is directed towards a fixed point and is proportional to its displacement from that point. SHM is an example of periodic motion.

Examples of Simple harmonic motion are

• The motion of the pistons in a gasoline engine
• Motion of a loaded test-tube

• Motion of mass suspended from a spring

• Motion of a simple pendulum

Basic terms used in simple harmonic motion

• Amplitude(A): This is the maximum displacement of the body from its equilibrium position. It is measured in metres
• Frequency(F): This is the number of oscillation the body makes in a second. Frequency is also the inverse of period(T). i.e F= 1/T, F = n/t

Where n is the number of complete oscillation.

Frequency is also called Cycle per seconds or revolution per second (rev/s)

Note: Cycle = revolution = oscillation

The unit of frequency is Hertz.

• Period(T): This is the time taken for the body to make a complete oscillation. It is the inverse of frequency. i.e T= 1/F, T= t/n

The unit of period is seconds.

The period of mass on a spring is 

Where m is the mass of the load in kg, k is the spring constant in N/m.

The period(T) of a simple pendulum is 

Where L is the length of the pendulum in metres and g is acceleration due to gravity.

The period of simple pendulum depends on the following

• Acceleration due to gravity (g)
• Length of the pendulum: The period is directly proportional to the square root of the length i.e

Let's take questions on all these formulae before we move to the next phase.

• A simple pendulum has a period of 17.0s. When the length is shortened by 1.5m, its period is 8.5s. Calculate the original length of the pendulum.

Solution

T1=17sec, L1=L, L2= L-1.5, T2= 8.5sec

• A simple pendulum of length 0.4m has a period 2s.What is the period of a similar pendulum of length 0.8m at the same place?

Solution

L1=0.4m, T1= 2sec, T2= ?, L2= 0.8m

• A simple pendulum makes 50 oscillations in one minute. Determine its period and frequency of oscillation

Solution

n= 50, t= 1min= 60secs, T=?, F= ?

• The period of a simple pendulum X is 5sec. What is the period of a simple pendulum Y which makes 50 vibrations in the same time it takes X to make 20 vibrations?

Solution

Tx = 5sec, nx= 20, ny= 50, Ty=?

• A student found out from a simple pendulum experiment that 20 oscillation were completed in 38sec. What is the period of oscillation of the pendulum?

Solution

n= 20, t= 38sec, T=?

• A boy timed 20 oscillations of a certain pendulum three times and obtained 44.3s, 45.5s and 45.7s respectively. Calculate the mean period of oscillation of the pendulum.

Solution

• In a simple pendulum experiment, a student observed that the times for 50 oscillations are 99.0, 99.5, 100.5 and 101.0s respectively. Calculate the mean period of oscillation of the pendulum

Solution

• A body of mass 0.02kg is suspended from the end of a spiral spring whose force constant is 0.4N/m. The body is set into a simple harmonic motion with an amplitude of 0.2m. Calculate the period and frequency.

Solution

m= 0.02kg, k= 0.4N/m, T=?, F=?

Let's calculate for period first, after which, we will now calculate for frequency

Relationship between angular displacement, angular velocity, frequency and period

Angular velocity (w) is the ratio of angular displacement(𑁜) to time i.e w= 𑁜/t, also note that w= 2πF, Since F=1/T, we can also say that w= 2π/T

Angular velocity is measured in radian per second (rad/s).

Let's take some questions on this formulae

• What is the angular velocity of a body whose frequency is 50Hz?

Solution

• Determine the period of oscillation of a body undergoing simple harmonic motion whose angular velocity is 628rad/s

 Solution

w= 628rad/s, T=?

Now  let's take a look at angular velocity, linear velocity, period and frequency, radius and amplitude. These are the formulae connecting them

Let's take some questions on this

• The period of oscillation of a particle executing simple harmonic motion is 4π seconds. If the amplitude of oscillation is 3.0m. calculate the maximum speed of the particle.

Solution

T=4π sec, A= 3.0m, V=?

• A body moves in SHM between two point 20m on the straight line Joining the points. If the angular speed of the body is 5 rad/s. Calculate its speed when it is 6m from the center of the motion.

Solution

w= 5 rad/s, x= 6m, to get amplitude, divide it by 20 by 2=10m

Let's take a look at angular acceleration, acceleration and angular velocity.

Angular acceleration is the ratio of angular velocity to time. This can be seen in equation 1. 

You can get questions on this under the topic Circular motion which has been treated.

Energy of SHM

Let's take a look at the motion of a simple pendulum

From the diagram above, it can be seen that at point B, the kinetic energy is maximum while at point A and C, the potential energy is maximum. But between point A and B, the body has both potential and kinetic energy. To learn more on kinetic and potential energy, click on energy

Formulae for calculating Energy of SHM

These formulae are used bases on the parameters given in a question.

Let's take an example

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From the diagram above, the bob of the pendulum has the fastest speed at?

Ans: X

Damped oscillation, Forced oscillation and Resonance

When a body is displaced through a small angle, it begins to perform SHM. During this period, it is observed that the amplitude reduces with time until it reduces entirely to zero. This type of oscillation or vibration is called Damped vibration

To maintain a constant amplitude, an external periodic force needs to be applied to the system. This type of oscillation or vibration resulting from an external periodic force is called Forced vibration.

Resonance is a phenomenon in which a vibrating body causes another body to  vibrate at its natural frequency.  Let's take a question on resonance.

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In the diagram above, which of the simple pendula will resonate with P when set into oscillation?

Note: The answer to this question will be the pendulum that has the same length as P

Ans: S

Exercises

1. Two simple pendulum X and Y make 400 and 500 oscillations respectively in equal time. If the period of oscillation of x is 1.5seconds, what is the period of oscillation of y?
2. A simple harmonic oscillator has a period of 0.02s and amplitude of 0.25m. Calculate the speed at the center of the oscillation
3. If a freely suspended object is pulled to one side and released, it oscillates about the point of suspension because the (a) velocity is minimum at the equilibrium point (b) acceleration is directly proportional to the square of the displacement (c) motion is directed away from the equilibrium point (d) acceleration is directly proportional to the displacement.
4. If the length of a simple pendulum is halved, its frequency is 

(a) increased by a factor of √2

 (b) increased by a factor of 2 

(c) decreased by a factor of 2

 (d) decreased by a factor of √2 

5)  

The diagram shows four positions of the bob of a simple pendulum. At which of these positions does the bob have maximum kinetic energy and minimum potential energy

(a) 1 (b) 2 (c) 3 (d) 4

Answer the questions and write your answers in the comment section below.