Displacement in Simple Harmonic Motion
The Oscillatory Motion has a big part to play in the world of Physics. Oscillatory motions are said to be harmonic if the displacement of the oscillatory body can be expressed as a function of sine or cosine of an angle depending upon time. In Harmonic Oscillations, the limits of oscillations on either side of the mean position may or may not be the same. Simple Harmonic Motion is a special kind of harmonic motion in which the limits of oscillation on either side of the mean position are the same.
Terms related to Simple Harmonic Motion:
1. Displacement (x): Displacement of a body executing Simple Harmonic Motion is defined as the net distance traveled by the body from its mean or equilibrium position.
x = A cos(ωt)
Considering the diametric projection of a body revolving in a circular path. A is the amplitude i.e. the maximum displacement from the mean position, ω is the angular velocity of the body revolving in the circular path, t is the time. Its S.I. unit is meter (m).
2. Amplitude (A): The maximum displacement of the body undergoing simple harmonic motion from the mean or equilibrium position is called the amplitude of oscillation.
3. Velocity (v): Velocity at any instant is defined as the rate of change of displacement with time.
v = dx/dt
For a body executing SHM, its velocity is maximum at the equilibrium position and minimum (zero) at the extreme positions where the value of displacement is maximum i.e. at x = A. It is a vector so it is important to mention both its value and direction for a complete explanation. The velocity of the body is inversely proportional to the force acting on the body, displacement from the mean position, and acceleration of the body. Its S.I unit is m/s.
4. Acceleration (a): Acceleration is defined as the rate of change of velocity with time. Acceleration is directly proportional to displacement and force. It is maximum at the extreme positions where the displacement is maximum (x = A) and minimum at the mean position (x = 0). The S.I unit of acceleration is m/s2.
5. Restoring Force (FR): Restoring Force is the force that always acts in a direction opposite to that of displacement from the mean position but is directly proportional to it. Restoring Force is maximum at the extreme positions and minimum at the mean position.
FR = -kx
where k is the spring constant and x is the displacement from the mean position. The S.I unit of Force is N or Kg m/s2.
6. Spring Constant (k): Spring Constant or Force Constant is a deterministic constant of a spring that determines the stiffness of the spring. It is the amount of work done to stretch or compress the spring by unit length. If the spring constant of a spring is 100 N/m then it means that 100 N of force is required to stretch or compress the spring by 1 m. The S.I unit of Spring Constant is N/m.
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. The Kinetic Energy of a body is due to its velocity and is given by,
K = (1/2) × m × v2
where m is the mass of the body and v is the velocity.
However, the Potential Energy of a body is by virtue of its position and is given by,
U = (1/2) × k × x2
where k is the spring constant and x is the displacement from the mean position. Therefore, the S.I Unit of Energy is kg m2/s2 or Joule.
8. Time Period (T): The Time Period of oscillation is defined as the time taken by the body to complete one full oscillation.
In other words, it is the time taken to cover 4 times the amplitude. The S.I unit of Time Period is seconds (s).
In the case of a simple pendulum, the formula for Time Period is given by,
T = 2π × √(l / g)
where T is the Time Period of oscillation, l is the length of the string or thread and g is the acceleration due to gravity.
9. Frequency (f): Frequency is defined as the total number of oscillations made by the body in one second. If the frequency of the body is 10 Hz, it means that the body completes 10 full oscillations in 1 second. Mathematically, it is defined as the reciprocal of the time period, i.e.
f = (1/T)
Therefore, the S.I unit of frequency is Hz or s-1
What is the Simple Harmonic Motion (SHM)?
The motion of a body is said to be simple harmonic if the velocity of the body at any instant is inversely proportional to the displacement from the mean position. Or in other words, the motion of a body is said to be simple harmonic if the restoring force acting on the body or the acceleration of the body at any instant is directly proportional to the displacement from the mean position and acts in a direction opposite to that of displacement.
SHM is a special kind of oscillatory motion in which the restoring force is directly proportional and opposite in direction to the displacement from the mean position.
Derivation of the equation for Displacement in SHM
The motion of a body moving in a circle with constant speed is called uniform circular motion. If we consider the motion of the body sideways, it looks as if the body is moving in a straight-line path (along the diameter). This to and fro motion of the body along the diameter is called simple harmonic motion.
Considering the above diagram, the motion of the body along the diameter (QR) is SHM. ‘O’ is the centre of the circle and in turn the equilibrium position of the system in SHM. x is the displacement from the mean position. ‘θ’ is the angular displacement.
Since, cos (θ) = (Base / Hypotenuse)
cos (θ) = x / AO
where AO is the radius and is equal to the maximum displacement from the mean position i.e. amplitude (A).
cos (θ) = x / A
x = A cos (θ) ……(1)
which is the equation of displacement from the mean position.
But since, angular displacement is the product of the angular velocity and the time taken by the particle.
θ = ω t
where ω is angular velocity and t is the time taken by the particle.
Now, equation (1) can be written as:
x = A cos (ω t) ……(2)
Also, ω = 2πf where f is the frequency of the particle.
x = A cos (2πf t) ……(3)
which is the equation of displacement of a body under SHM. The maximum displacement of a body under SHM is called amplitude and is denoted by ‘A’. With the help of the above equation (equation of motion or equation of displacement), we can find the equation for velocity and acceleration too.
Problem 1: Considering a body executing simple harmonic motion, find the equation of the Time Period in terms of displacement.
Considering the below diagram for SHM,
V = (2 × π × R) / T
where T is Time Period.
V = (2 × π × A) / T ……(1)
Here V = Vmax
(V is the velocity of the body moving in circular motion and Vmax is the maximum velocity of the body moving in SHM along the diameter of the circle)
Vmax = √(k / m) × A ……(2)
Putting the value of Vmax in equation (1) as,
√(k / m) × A = (2 × π × A) / T
T = 2 × π × √(m / k)
Now, k = FR / x, FR is the restoring force acting on the body at a displacement of x units from the mean position.
T = 2 × π × √((m × x) / FR)
Problem 2: Derive the equation for the instantaneous velocity of a body executing Simple Harmonic Motion.
Since, it is known that:
Total Energy = Kinetic Energy + Potential Energy.
(1/2) × k × A2 = (1/2) × m × v2 + (1/2) × k × x2
where k is spring constant, m is mass, x is displacement and v is the velocity.
K × A2 = m × v2 + k × x2
v = √((K × A2 – k × x2) / m)
= √(k / m) × A × √(1 – (x2 / A2))
where √(k / m) × A is the maximum velocity.
= vmax × √(1 – (x2 / A2))
Problem 3: Calculate the ratio of displacement to amplitude when the kinetic energy of a body is twice its potential energy.
K = 1/2 × m × v2
U = 1/2 × k × x2
Given that, K = 2 × U
(1/2) × m × v2 = 2 × (1/2) × k × x2 ……(1)
Instantaneous velocity is given by,
v = √(k / m) * A * √(1 – (x2 / A2))
where √(k / m) × A is the maximum velocity.
Putting the value of v in equation (1) we get,
A2 – x2 = 2 × x2
A2 = 3 × x2
x / A = 1 / √3
x : A = 1 : √3
Problem 4: The force acting on a body under SHM is 200 N. If the Spring constant is 50 N/m. Find the displacement from the mean position.
The formula to calculate the restoring force is,
FR= -k × x
Substituting the given values in the above equation as,
200 N = -(50 × x)
x = -4 m
where negative sign indicates that the force and the displacement are opposite in direction.
Problem 5: Derive an expression for the amount of work done or potential energy of a body executing SHM.
Since, the work done is defined as,
Work Done = Force × Displacement
But the formula cannot be used directly to find the work done or potential energy because force acting on a body
under SHM is not constant. Force is a function of x, FR = -kx.
The formula for work done in the case of SHM :
W = (1/2) × k × x2 [ For mass-spring system, where k is spring constant and x is the displacement from the mean position.]
For variable force,
W = ∫ F dx
W = ∫ kx dx [F = kx]
W = (1/2) × k × x2
This is the formula for the work done or the potential energy of the mass-spring system denoted by U.