Question

Distinguish between an ammeter and a voltmeter. (Two points each).
The displacement of a particle performing simple harmonic motion is \( \frac{1}{3} \) of its amplitude. What fraction of total energy is its kinetic energy?
 

Hint:

In simple harmonic motion, the total energy is the sum of kinetic and potential energies. The kinetic energy depends on the displacement of the particle, and the fraction of the total energy that is kinetic can be determined based on the displacement.

Answer 1

An ammeter and a voltmeter are both instruments used to measure electrical quantities, but they serve different purposes and have distinct characteristics. The differences between an ammeter and a voltmeter are as follows: 
Ammeter: An ammeter measures the current flowing through a circuit. It is always connected in series with the circuit components. It has very low resistance to ensure it does not impact the current in the circuit.
Voltmeter: A voltmeter measures the potential difference (voltage) between two points in a circuit. It is always connected in parallel across the components. It has very high resistance to prevent it from drawing any current from the circuit. 

The total energy \( E \) of a particle undergoing simple harmonic motion is the sum of its kinetic energy \( K \) and potential energy \( U \). The total energy remains constant and is given by the formula: \[ E = \frac{1}{2} m \omega^2 A^2 \] where \( m \) is the mass of the particle, \( \omega \) is the angular frequency, and \( A \) is the amplitude. The displacement \( x \) of the particle is related to its amplitude \( A \) as: \[ x = \frac{A}{3} \] The potential energy at displacement \( x \) is given by: \[ U = \frac{1}{2} m \omega^2 x^2 \] Substituting \( x = \frac{A}{3} \): \[ U = \frac{1}{2} m \omega^2 \left(\frac{A}{3}\right)^2 = \frac{1}{2} m \omega^2 \frac{A^2}{9} \] Now, the kinetic energy \( K \) is the difference between the total energy and the potential energy: \[ K = E - U = \frac{1}{2} m \omega^2 A^2 - \frac{1}{2} m \omega^2 \frac{A^2}{9} \] \[ K = \frac{1}{2} m \omega^2 A^2 \left(1 - \frac{1}{9}\right) \] \[ K = \frac{1}{2} m \omega^2 A^2 \left(\frac{8}{9}\right) \] Thus, the fraction of the total energy that is kinetic energy is: \[ \frac{K}{E} = \frac{\frac{8}{9}}{1} = \frac{8}{9} \] Therefore, the fraction of the total energy that is kinetic energy is \( \frac{8}{9} \). \bigskip