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Force Law for Simple Harmonic Motion

  • Difficulty Level : Expert
  • Last Updated : 22 Jul, 2021
Geek Week

Have you ever wondered why, when we stretch an elastic band and then let it go, it returns to its previous state? It is compelled to revert to its original state by a force. But what exactly is this force? Let us investigate this force and develop the force law for simple harmonic motion. 

Periodic Motion is something we’re already familiar with. Periodic motion is defined as motion that repeats itself at equal intervals of time. For example, the motion of a clock’s hands, the motion of a car’s wheels, and the motion of a merry-go-round. In nature, all of these motions are repeated. They repeat themselves after a certain period of time.

An Oscillatory Motion is a periodic movement in which an item oscillates about its equilibrium position. After a given amount of time, the item repeats the same sequence of moves. An Oscillation is one such series of motions. An oscillatory motion may be seen in the movement of a basic pendulum, the movement of leaves in a breeze, and the movement of a cradle.

Simple Harmonic Motion (SHM)

Simple harmonic motion is the most basic type of oscillatory motion. When an object moves in a straight path, it exhibits simple harmonic motion. All oscillatory motion examples are instances of basic harmonic motion. 

Swinging a basic pendulum causes it to move away from its mean equilibrium point. When it reaches its extreme position, where it has the greatest displacement, it comes to a halt, and its velocity becomes zero. It returns to its equilibrium position as a result of a force acting in the direction of the equilibrium position.



It now travels through its normal location but does not stop. It shifts to its other extreme position. After that, it returns to its original place. An Oscillation is a type of full motion. A basic pendulum’s swing is an excellent illustration of simple harmonic motion.

Thus, the motion of a body is said to be simply harmonic if the restoring force acting on it is directly proportional to the displacement from the mean position and always tends to oppose it. The direction of the restoring force is opposite to the direction of displacement.

The acceleration of a particle moving in a simple harmonic motion is given by,

a(t) = -ω2x(t)

where ω is the angular velocity of the particle.

Let’s now discuss some important terms related to a Simple Harmonic Motion of a particle as

  1. Displacement (x): Displacement at any instant of time is defined as the net distance travelled by the body executing SHM from its mean or equilibrium position.
  2. Amplitude (A): The amplitude of oscillation is defined as the maximum displacement of the body executing SHM on either side of the mean position.
  3. Velocity (v): Velocity at any instant is defined as the rate of change of displacement with time. For a body executing SHM, its velocity is maximum at the mean position and minimum (zero) at the extremes. The Velocity of the body is inversely proportional to the displacement from the mean position.
  4. Acceleration (a): Acceleration is defined as the rate of change of velocity with time. Unlike velocity, acceleration is directly proportional to displacement. It is maximum at the extreme positions where the displacement is maximum and minimum at the mean position (displacement = 0).
  5. Restoring Force (FR): Restoring Force is the force that always acts in a direction opposite to that of displacement but is directly proportional to it. Restoring Force is maximum at the extreme positions and minimum at the mean position.
  6. Spring Constant (k): Spring Constant is a constant value for a particular spring that determines the amount of force required to compress or stretch the spring by 1 unit.
  7. Energy (E): The total energy of the body under SHM is called mechanical energy, mechanical energy of the body remains constant throughout the motion if the medium is frictionless. The Mechanical Energy of a body at any instant is the sum total of its kinetic and potential energy.
  8. Time Period (T): The Time Period of oscillation is defined as the time taken by the body to complete one oscillation. In other words, it is the time taken to cover 4 times the amplitude.
  9. Frequency (f): Frequency is defined as the number of oscillations made by the body in one second. It is reciprocal of the time period. f = (1/T)

Note: In periodic motion, the direction of restoring force may or may not be in the direction of displacement but in Simple Harmonic Motion (SHM) the direction of restoring force is always opposite to the direction of displacement. This leads to the fact that all Simple Harmonic Motions are periodic motions but vice versa is not true.

Force Law For Simple Harmonic Motion

Let us use an example to develop the force law for simple harmonic motion. The most basic example of the simple harmonic motion is a spring-block system. Consider a mass m block attached to a spring, which is then attached to a stiff wall. The block is supported by a frictionless surface.



The spring is at its equilibrium position when we do not pull it, that is when no force is exerted to it. The net force acting on it is zero in this condition. Let’s try two different things and see what happens.

  1. When we move the block outwards, a force acts on it, attempting to draw it inwards, towards its equilibrium position.
  2. When we press the block inwards, a force operating on it tries to push it outwards, towards its equilibrium position.

In both situations, we can observe that a force is operating on the block to try to return it to its equilibrium position. This force is the restoring force, and it is the foundation of the force law for simple harmonic motion. Let’s figure out how to apply this concept. 

Variation in restoring force with change in displacement

Let F be the restoring force and x denote the displacement of the block from its equilibrium position. As a result of our observations, we may conclude that the restoring force is directly proportional to the displacement from the mean position.

This implies, 

F = – kx                                                                                                                                                                                                    ……(1)

where k is known as the force constant. It is measured in N/m in the SI system and in dynes/cm in the C.G.S. system. The negative symbol denotes that the restoring force and displacement are always pointing in opposing directions. For simple harmonic motion, equation (1) is the simplest version of the force law.

It demonstrates the fundamental law of simple harmonic motion, which states that force and displacement must be in opposing directions.

We also know that: 

F = ma

As a result, 



a = F/m

Substituting the value of F from equation (1) which yields,

a = – kx/m = – ω2x                                                                                    (where k/m = ω2)                                                               ……(2)

As a result, Equations I and II are the force laws of simple harmonic motion. It should be noted that the restoring force is always directed towards the mean position and in the opposite direction of displacement.

Sample Problems

Problem 1: The amount of force required to stretch a spring by 10 cm is 150 N. How much force is required to stretch the spring by 100 cm? Calculate the spring constant of the spring.

Solution:

As long as the external force is applied on the body, the restoring and the applied force are the same, then

F = -kx

Since, it is given that, F = 150 N

x = -10 cm = – 0.1 m (Taking x negative because we have taken force as positive)



Therefore, k = (150 / 0.1) = 1500 N/m

Now, x = -100 cm = -1 m

F = -kx

F = -(1500) × -1

F = 1500 N

Thus, Spring Constant = 1500 N/m

Problem 2: Calculate the amount of work done to compress a spring having a spring constant of 1000 N/m by 30 m.

Solution:

Work done = (1/2) × k × x2

                   = (1/2) × 1000 × (30)2 Nm or Joule

                   = 500 × 900 Joule

                   = 450000 Joule         

Problem 3: A body is moving in a circular motion having a Time Period equal to 10 seconds. Considering the motion of the body along the diameter of the circular path. If the force acting on the body at a displacement from the mean position is 200 N then find the acceleration at that point. (Consider the diameter to be made of spring having spring constant = 100 N/m)

Solution:  

Given that, T = 10 seconds

According to Hooke’s law, F = -kx

200 = -(100) * x

x = -2 m [Negative sign indicates that the direction of displacement is opposite to that of force]

Acceleration = -ω2x

ω = 2πf, 

where f is the frequency of oscillation given by, 

f =1/T

f = (1/10) Hz, 

And

ω = 2 × 3.14 × 0.1

    = 0.628 radian/second

Acceleration = -(0.628)2 × (-2) m/s2

                          = 0.788768 m/s2

Problem 4: A body having a mass of 10 Kg has a velocity of 3 m/s after 2 seconds of its staring from the maximum displacement position. If the frequency is (1/8) Hz, find the potential energy and kinetic energy of the body at that point and even find the total energy.    

Solution:   



Since, K = (1/2) × m × v2

                    = (1/2) × 10 × 9 J

                    = 45 J

  ω = 2πf

      = (2 × 3.14) / 8

      = 0.785 radians/second

Instantaneous velocity is given by,

v = A × ω × sin (ωt)                                                                                                  (Neglecting the negative sign)

3 = A × (0.785) × sin((π/4) × 2)),  

0.785 = (π/4) (approx)

or

A = 3 / 0.785 m

A = 3.8217 m

Total Energy(E) = (1/2) * m * ω2 * A

                         = (1/2) * 10 * (0.785)2 * (3.8217)2

                         = 45 J

Potential Energy(U) = Total Energy(E) – Kinetic Energy(K)

                                = 45 – 45 J

                                 = 0 J

This indicates that the body is at the mean position.

Problem 5: What is the significance of the Spring Constant? Write the dimension of the Spring Constant.

Solution:

Spring constant of a spring is the amount of force required to stretch or compress the given spring by unit displacement. It is used to compare the stiffness of two springs. The one with greater spring constant has more stiffness i.e. is more difficult to deform compared to the one having lower spring constant. 

In other words, the amount of work done to stretch or compress a spring (mass-spring system) is directly proportional to its spring constant. Higher the spring constant, higher the amount of work done to stretch or compress it. Unit of Spring Constant is N/m. And the Dimension of Spring Constant is [M L0 T-2].

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