In a simple harmonic motion, let f be the acceleration and T be the time period. If x denotes the displacement, then |fT| vs. x graph will look like:
The Correct Option is B
Solution and Explanation
Understanding the |fT| vs. x graph in Simple Harmonic Motion (SHM):
In SHM, the total force \( f_T \) acting on a particle is directly proportional to the displacement \( x \) from the mean position, and always directed towards the mean position:
\[ f_T = -kx \]
Here, \( k \) is the force constant. Taking the magnitude, we get:
\[ |f_T| = k|x| \]
This means the graph of \( |f_T| \) versus \( x \) is a V-shaped graph—linear on both sides of the origin, with the minimum value (zero) at \( x = 0 \) and increasing symmetrically on either side as \( |x| \) increases.
Important observations:
- At \( x = 0 \), \( |f_T| = 0 \) (equilibrium position)
- As \( x \) increases (either positive or negative), \( |f_T| \) increases linearly
- The graph is symmetric about the origin
Thus, the correct shape of the graph is a straight V-line centered at the origin, and the correct option is (B).
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Concepts Used:
Simple Harmonic Motion
Simple Harmonic Motion is one of the most simple forms of oscillatory motion that occurs frequently in nature. The quantity of force acting on a particle in SHM is exactly proportional to the displacement of the particle from the equilibrium location. It is given by F = -kx, where k is the force constant and the negative sign indicates that force resists growth in x.
This force is known as the restoring force, and it pulls the particle back to its equilibrium position as opposing displacement increases. N/m is the SI unit of Force.
Types of Simple Harmonic Motion
Linear Simple Harmonic Motion:
When a particle moves to and fro about a fixed point (called equilibrium position) along with a straight line then its motion is called linear Simple Harmonic Motion. For Example spring-mass system
Conditions:
The restoring force or acceleration acting on the particle should always be proportional to the displacement of the particle and directed towards the equilibrium position.
- – displacement of particle from equilibrium position.
- – Restoring force
- - acceleration
Angular Simple Harmonic Motion:
When a system oscillates angular long with respect to a fixed axis then its motion is called angular simple harmonic motion.
Conditions:
The restoring torque (or) Angular acceleration acting on the particle should always be proportional to the angular displacement of the particle and directed towards the equilibrium position.
Τ ∝ θ or α ∝ θ
Where,
- Τ – Torque
- α angular acceleration
- θ – angular displacement