An insect of mass 20g crawls from the centre to the outside edge of a rotating disc of mass 200g & radius 20cm. The disc was initially rotating at 22.0 rad/s. What will be it's final angular velocity? What is the kinetic energy in the system?
To find the insect's final angular velocity and the kinetic energy in the system, we can apply the principle of conservation of angular momentum and the principle of conservation of energy.
1. Final Angular Velocity:
The principle of conservation of angular momentum states that the total angular momentum before an event is equal to the total angular momentum after the event, assuming no external torques act.
The formula for angular momentum is given by:
Angular Momentum (L) = moment of inertia (I) * angular velocity (ω)
To apply conservation of angular momentum, we need to calculate the initial and final angular momenta:
Initial Angular Momentum (L1) = Moment of inertia of the disc (I1) * Initial angular velocity of the disc (ω1)
Final Angular Momentum (L2) = Moment of inertia of the disc with insect (I2) * Final angular velocity of the disc with insect (ω2)
The moment of inertia of a disc is given by:
I = (1/2) * mass * radius^2
Let's calculate the initial and final angular momenta:
Moment of inertia of the disc (I1) = (1/2) * 0.2 kg * (0.2 m)^2
Moment of inertia of the disc with insect (I2) = (1/2) * (0.2 kg + 0.02 kg) * (0.2 m)^2
= (1/2) * 0.22 kg * (0.2 m)^2
Using the principle of conservation of angular momentum:
L1 = L2
(I1 * ω1) = (I2 * ω2)
Solving for ω2, we have:
ω2 = (I1 * ω1) / I2
Substituting the values, we get:
ω2 = [(1/2) * 0.2 kg * (0.2 m)^2 * 22.0 rad/s] / [(1/2) * 0.22 kg * (0.2 m)^2]
Simplifying, we find:
ω2 = 20.0 rad/s
Therefore, the insect's final angular velocity is 20.0 rad/s.
2. Kinetic Energy in the System:
The kinetic energy of the system is the sum of the kinetic energy of the disc and the kinetic energy of the insect.
The kinetic energy of an object is given by:
Kinetic Energy (KE) = (1/2) * mass * velocity^2
The velocity of a point on the edge of a rotating disc is given by:
Velocity = radius * angular velocity
Let's calculate the kinetic energy of the disc and the insect:
Kinetic Energy of the disc (KE_disc) = (1/2) * 0.2 kg * (0.2 m)^2 * (22.0 rad/s)^2
Kinetic Energy of the insect (KE_insect) = (1/2) * 0.02 kg * (0.2 m * 22.0 rad/s)^2
Total Kinetic Energy (KE_total) = KE_disc + KE_insect
Substituting the values, we find:
KE_total = [(1/2) * 0.2 kg * (0.2 m)^2 * (22.0 rad/s)^2] + [(1/2) * 0.02 kg * (0.2 m * 22.0 rad/s)^2]
Simplifying, we get:
KE_total ≈ 0.1216 Joules
Therefore, the kinetic energy in the system is approximately 0.1216 Joules.