The velocity of the center of mass of the rod is found by dividing the impulse by the rod's mass, and the angular velocity is calculated based on the impulse, the rod's mass, and its length.
The student's question is about finding the velocity of the center of mass and the angular velocity of a uniform rod just after an impulse is applied at one end perpendicular to the length of the rod. Applying the principle of conservation of linear momentum, the velocity of the center of mass can be calculated using the formula:
Vcm = P / m
where Vcm is the velocity of the center of mass, P is the impulse applied, and m is the mass of the rod. To find the angular velocity, we need to consider the moment of inertia (I) of the rod about the center of mass and the torque (τ) caused by the impulse. The angular velocity (ω) can be determined by:
ω = τ / I
The torque τ can be expressed as the product of the impulse and the perpendicular distance from the point of application of the impulse to the axis of rotation, which is L/2. The moment of inertia I for a rod revolving around its center is given by (1/12) mL². Substituting these values gives us:
ω = (P * (L/2)) / ((1/12) * m * L²).
By simplifying the equation, we obtain the final formula for ω.