A turntable of radius 2m rotates freely about a fixed vertical axis. A child 19 kg is standing on the turntable right at the outer edge and the system is initially rotating at an angular speed of 0.5 rev/s. The child then moves to position 1m from center of the turntable. what does the urntable angular speed become? The moment of inertia is 400kgm^2.
conservation of momentum
29*2^2*.5+400*.5=29*1^2*w + 400*w
solve for w, in revs/sec
I don't really understand we use I1w1=I2w2?
but according to your equation its
mr^2+Iw = mr^2+Iw ... which is equivalent to I+L = I+L <-- i don't get this part?
and the answer is 0.568 rev/s which according to your way is right but its 19 instead of 29.
i mean..
mr^2w+Iw = mr^2w+Iw ... which is equivalent to L+L = L+L
The moment of inertialfor the kid is mass*distance^2.
sorry about the 29.
ohh okies thanks for clearing that up (:
To determine the angular speed of the turntable after the child moves towards the center, we need to apply the principle of conservation of angular momentum.
Angular momentum is the product of moment of inertia (I) and angular velocity (ω). In this case, the angular momentum is conserved because there are no external torques acting on the system.
Before the child moves, the initial angular momentum (L1) can be calculated by multiplying the moment of inertia (I) by the initial angular speed (ω1):
L1 = I * ω1
L1 = 400 kgm^2 * (0.5 rev/s) (1 revolution = 2π radians)
L1 = 400 kgm^2 * (0.5 * 2π rad/s)
L1 = 400 kgm^2 * (π rad/s)
L1 = 400π kgm^2/s
When the child moves towards the center, the moment of inertia of the system changes, but the angular momentum remains conserved.
After the child moves, the final angular speed (ω2) can be calculated by dividing the initial angular momentum (L1) by the new moment of inertia (I2):
L1 = I2 * ω2
ω2 = L1 / I2
To find I2, we can apply the law of conservation of angular momentum. The initial moment of inertia (I1) can be calculated using the formula for a uniform disk:
I1 = 0.5 * m * r1^2
I1 = 0.5 * 19 kg * (2m)^2
I1 = 0.5 * 19 kg * 4 m^2
I1 = 38 kgm^2
The final moment of inertia (I2) after the child moves can be calculated using the same formula, but with the new radius (r2 = 1m):
I2 = 0.5 * m * r2^2
I2 = 0.5 * 19 kg * (1m)^2
I2 = 0.5 * 19 kg * 1 m^2
I2 = 9.5 kgm^2
With the values calculated, we can now determine the final angular speed (ω2):
ω2 = L1 / I2
ω2 = (400π kgm^2/s) / (9.5 kgm^2)
ω2 ≈ 42.105 rad/s
To convert the angular speed from radians per second to revolutions per second, we can use the conversion factor:
1 revolution = 2π radians
Therefore:
ω2 ≈ 42.105 rad/s * (1 rev / 2π rad)
ω2 ≈ 6.7 rev/s
Hence, the angular speed of the turntable after the child moves 1m from the center is approximately 6.7 rev/s.