A woman with mass of 63 kg stands at the
rim of a horizontal table having a moment of
inertia of 520 kg m2
and a radius of 1.5 m.
The turntable is initially at rest and is free
to rotate about a frictionless, vertical axis
through its center. The woman then starts
walking around the rim clockwise (as viewed
from above the system) at a constant speed of
1.3 m/s relative to the Earth.
With what angular speed does the
turntable rotate?
Answer in units of rad/s
To find the angular speed at which the turntable rotates, we can use the principle of conservation of angular momentum.
Angular momentum is the product of moment of inertia and angular velocity:
L = Iω
Where:
L is the angular momentum
I is the moment of inertia
ω is the angular velocity (angular speed)
Initially, the turntable is at rest, so the angular momentum is zero. When the woman starts walking around the rim, she adds angular momentum to the system.
The angular momentum of the woman can be calculated using her mass, radius, and velocity:
L_woman = (m_woman * r_woman) * v_woman
Where:
m_woman is the mass of the woman
r_woman is the radius from the axis of rotation to the woman (equals the radius of the turntable)
v_woman is the velocity of the woman
Since the total angular momentum is conserved, we can set the initial angular momentum to be equal to the angular momentum after the woman starts walking:
L_initial = L_woman
I_initial * ω_initial = (m_woman * r_woman) * v_woman
Solving for the angular velocity ω_initial:
ω_initial = [(m_woman * r_woman) * v_woman] / I_initial
Now we can substitute the given values:
m_woman = 63 kg
r_woman = 1.5 m
v_woman = 1.3 m/s
I_initial = 520 kg m^2 (moment of inertia of the table)
Plugging in these values:
ω_initial = [(63 kg * 1.5 m) * 1.3 m/s] / 520 kg m^2
ω_initial = 2.01 rad/s
Therefore, the initial angular speed of the turntable is 2.01 rad/s.