A woman with mass of 57 kg stands at the rim of a horizontal table having a moment of inertia of 570 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 1.9 m/s relative to the Earth.With what angular speed does the turntable rotate?
Answer in units of rad/s.
I omega-woman = I omega-turntable
Treat woman as point mass I = mr^2 and her omega is v/r
To determine the angular speed with which the turntable rotates, we can use the principle of conservation of angular momentum.
The angular momentum of the woman and the turntable system is given by the equation:
L_initial = I * ω_initial
where L_initial is the initial angular momentum, I is the moment of inertia of the turntable, and ω_initial is the initial angular speed of the turntable.
Initially, the turntable is at rest, so its initial angular speed is 0.
We can calculate the angular momentum of the woman using the equation:
L_woman = m * r * v
where m is the mass of the woman, r is the radius of the turntable, and v is the speed of the woman relative to the Earth.
Substituting the given values:
L_woman = (57 kg) * (1.5 m) * (1.9 m/s)
= 154.35 kg·m²/s
Since the total angular momentum is conserved, we can set L_initial equal to L_woman:
I * ω_initial = L_woman
Rearranging the equation to solve for ω_initial:
ω_initial = L_woman / I
Substituting the given values:
ω_initial = 154.35 kg·m²/s / 570 kg·m²
= 0.270 rad/s
Therefore, the angular speed with which the turntable rotates is approximately 0.270 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 given by the equation:
Angular momentum (L) = moment of inertia (I) × angular speed (ω)
The initial angular momentum of the system is zero since the turntable is at rest initially. After the woman starts walking, the system gains angular momentum due to the woman's motion.
The angular momentum of the woman can be calculated as follows:
Angular momentum of the woman (Lw) = mass of the woman (mw) × velocity of the woman (vw) × radius of the table (r)
Given that the woman's mass (mw) is 57 kg, velocity (vw) is 1.9 m/s, and the radius (r) is 1.5 m, we can calculate Lw:
Lw = 57 kg × 1.9 m/s × 1.5 m
Next, we equate the initial and final angular momenta and solve for the angular velocity (ω) of the turntable:
0 = Lw + (I × ω)
Rearranging the equation, we find:
ω = -Lw / I
Substituting the known values, we have:
ω = -(57 kg × 1.9 m/s × 1.5 m) / 570 kg m²
Calculating this expression, we find the angular speed (ω) of the turntable.
Please note that the negative sign indicates that the turntable rotates in the opposite direction to the woman's motion.