A phonograph record of radius 6 cm and mass 15 g is rotating with an angular velocity of 33.33 rev/min. A Penny of mass 1 g suddenly falls on the record and lands at the radial distance of 4 cm from the record’s center. What is the angular velocity after the collision.
To determine the angular velocity after the collision, we need to apply the law of conservation of angular momentum.
The angular momentum of an object is given by the formula L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
Before the collision, the phonograph record is rotating with the initial angular velocity of 33.33 rev/min. The moment of inertia of the record can be calculated using the formula:
I = 0.5 * M * R^2,
where M is the mass of the record and R is the radius. Plugging in the values, we have:
I = 0.5 * 0.015 kg * (0.06 m)^2 = 5.4 x 10^-5 kg·m^2.
The initial angular momentum is:
L_initial = I * ω_initial = 5.4 x 10^-5 kg·m^2 * (33.33 rev/min * 2π/rev * 1 min/60 s) = 0.034 kg·m^2/s.
Now, let's consider the moment of inertia after the penny falls on the record. Since the penny lands at a 4 cm radial distance from the center, we can calculate its moment of inertia using the formula:
I_penny = m * r^2,
where m is the mass of the penny and r is the radial distance. Plugging in the values, we have:
I_penny = 0.001 kg * (0.04 m)^2 = 1.6 x 10^-6 kg·m^2.
The total moment of inertia after the collision is the sum of the moment of inertia of the record and the penny:
I_total = I + I_penny = 5.4 x 10^-5 kg·m^2 + 1.6 x 10^-6 kg·m^2 = 5.56 x 10^-5 kg·m^2.
Now, to find the angular velocity after the collision, we rearrange the formula for angular momentum:
L_final = I_total * ω_final,
and solve for ω_final:
ω_final = L_final / I_total.
Since angular momentum is conserved, we have L_initial = L_final, so:
ω_final = L_initial / I_total = 0.034 kg·m^2/s / 5.56 x 10^-5 kg·m^2 = 611.51 rad/s.
Therefore, the angular velocity after the collision is approximately 611.51 rad/s.