Suppose a 69 kg person stands at the edge of a 7.8 m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1600 kg*m^2. The turntable is at rest initially, but when the person begins running at a speed of 3.7 m/s (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.
momentum conservation
I1*W1= I2*w2
Wturntable=(massman*r^2)3.7(2PI/r)/1600
To solve this problem, we can apply the principle of conservation of angular momentum.
The formula for angular momentum is given by:
L = I * ω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
Initially, the turntable and the person are at rest, so the total angular momentum is zero:
L(initial) = 0
When the person starts running, the person's angular momentum is given by:
L(person) = m * r * v
Where m is the mass of the person, r is the radius of the turntable, and v is the speed of the person with respect to the turntable.
In this case, the person's angular momentum is in the clockwise direction, and since the turntable starts rotating in the opposite direction, the total angular momentum after the person starts running is given by:
L(final) = -L(person)
Since angular momentum is conserved, we can equate the initial and final angular momentum:
L(initial) = L(final)
0 = -L(person)
Substituting in the values, we get:
0 = -m * r * v
Now we can solve for the angular velocity of the turntable:
L = I * ω
0 = I * ω
0 = (m * r^2) * ω
Solving for ω, we get:
ω = 0 / (m * r^2)
ω = 0
Therefore, the angular velocity of the turntable is 0 rad/s. The turntable does not rotate in the opposite direction when the person starts running.