Find the height of a conical pendulum if the time period is given.
Solution and Explanation
If we have the length of the string and the time period, we can calculate the height using the following steps:
1. Identify the length of the string in meters.
2. Calculate the time period of the conical pendulum in seconds.
3. Determine the value of gravitational acceleration, typically taken as 9.8 m/s².
4. Use the formula for the time period of a conical pendulum:
\(T = 2\pi\sqrt{\frac{h}{g}}\)
Where T is the time period, h is the height, and g is the gravitational acceleration.
5. Rearrange the formula to solve for the height:
\(h = \left(\frac{T}{2\pi}\right)^2 \times g\)
6. Substitute the known values of T and g into the formula and calculate the height.
Please note that this calculation assumes a perfectly ideal conical pendulum and neglects factors like air resistance and the mass of the string.
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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