Question

A particle performing uniform circular motion has angular momentum \(L\)., its angular frequency is doubled and its \(K . E\). halved, then the new angular momentum is \(\\{\mathrm{A}\\} 1 / 2\) \\{B \(\\} 1 / 4\) \(\\{\mathrm{C}\\} 2 \mathrm{~L}\) \(\\{\mathrm{D}\\} 4 \mathrm{~L}\)

Short answer

The new angular momentum is \(L' = \frac{1}{2} L\), which corresponds to option A.

Step-by-step solution

Define given variables

Initially, Angular momentum L Angular frequency ω Kinetic energy K.E After the changes, New angular frequency = 2ω New kinetic energy = (1/2) K.E

Introduce the formulas for angular momentum and kinetic energy

The angular momentum formula is: \(L = Iω\), where I is the moment of inertia and ω is the angular frequency. The kinetic energy formula for rotational motion is: \(K.E = \frac{1}{2} Iω^2\)

Express the moment of inertia in terms of angular momentum and angular frequency

Using the angular momentum formula, we can express moment of inertia as: \(I = \frac{L}{ω}\)

Express the initial kinetic energy in terms of the moment of inertia and angular frequency

Using the moment of inertia expression from step 3 and the kinetic energy formula: \(K.E = \frac{1}{2}(\frac{L}{ω})ω^2 = \frac{1}{2}Lω\)

Calculate the new angular momentum

The new kinetic energy is half of the initial, so: \(\frac{1}{2}K.E = \frac{1}{2}(\frac{1}{2}Lω) = \frac{1}{4}Lω\) Since the new angular frequency is 2ω: \(\frac{1}{2}K.E = \frac{1}{4}L(2ω) = \frac{1}{2}L(ω)\) New angular momentum \(L' = \frac{1}{2} L\) The new angular momentum is \(L' = \frac{1}{2} L\), which corresponds to option A.