A large horizontal circular platform (M=143.1 kg, r=3.01 m) rotates about a frictionless vertical axle. A student (m=75.3 kg) walks slowly from the rim of the platform toward the center. The angular velocity ω of the system is 2.10 rad/s when the student is at the rim. Find ω (in rad/s) when the student is 2.89 m from the center.
Ok,
Original I = (1/2)(143.1)(3.01^2 + 75.3*(3.01)^26
= 648 + 682 = 1330 kg m^2
Final I = 648 + 75.3(2.89)^2 = 1276
1330 (2.1) = 1276 omega
so
omega = 2.19 radians/second
conservation of angular momentum (the figure skater spins faster when she pulls her arms in
I1 omega1 = I2 omega2
I of platform = (1/2)(143.1)(3.01)^2
original I of student = m r^2 = 75.3(3.01)^2
final I of student = 75.3(2.89)^2
something is wrong in the answer, can you explain it again please?..
Thank you very much.
Ah, you've got yourself quite the rotating situation! Let's clown around with some physics here.
We can start by applying the principle of conservation of angular momentum. In this case, since there are no external torques acting on the system, the initial angular momentum of the system will be equal to the final angular momentum of the system.
The initial angular momentum L_initial can be calculated as the product of the moment of inertia and the initial angular velocity:
L_initial = I_initial * ω_initial
Similarly, the final angular momentum L_final can be calculated using the moment of inertia and the final angular velocity:
L_final = I_final * ω_final
Since angular momentum is conserved, L_initial = L_final. Therefore, we have the equation:
I_initial * ω_initial = I_final * ω_final
Now, let's find the initial and final moments of inertia, denoted as I_initial and I_final. For a uniform circular platform, the moment of inertia can be calculated using the formula:
I = (1/2) * M * r^2
So, for the initial moment of inertia, when the student is at the rim, we have:
I_initial = (1/2) * M * r^2
For the final moment of inertia, when the student is 2.89 m from the center, we need to consider the part of the platform still remaining (r_remaining) and the part walked by the student (r_student):
I_final = (1/2) * M * r_remaining^2 + m * r_student^2
Given that the platform radius is 3.01 m, the remaining radius can be calculated as:
r_remaining = 3.01 m - 2.89 m = 0.12 m
With all these juggling steps, we can now set up a ratio equating the initial angular velocity ω_initial to the final angular velocity ω_final:
(I_initial * ω_initial) / (I_final) = ω_final
Plugging in the values of I_initial, I_final, and ω_initial:
[(1/2) * M * r^2 * ω_initial] / [(1/2) * M * r_remaining^2 + m * r_student^2] = ω_final
Simplifying the equation by canceling out some terms and plugging in the known values, we'll eventually arrive at the answer. Just keep in mind that I can be the life of the party, but I'll leave the final calculation to you!
To find the angular velocity (ω) when the student is 2.89 m from the center, we can use the principle of conservation of angular momentum.
The formula for angular momentum is given by L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
For a rotating system, the moment of inertia is given by I = MR^2, where M is the mass of the platform and R is the distance from the axis of rotation (in this case, the center of the platform).
Initially, when the student is at the rim, the angular momentum of the system is L_initial = I_initial * ω_initial. We can calculate I_initial using the formula I_initial = M * R_initial^2.
When the student moves towards the center and reaches a distance of 2.89 m, the angular momentum of the system is L_final = I_final * ω_final. We can calculate I_final using the formula I_final = M * R_final^2.
According to the conservation of angular momentum, L_initial = L_final. Therefore, I_initial * ω_initial = I_final * ω_final.
To find ω_final, we rearrange the equation as ω_final = (I_initial * ω_initial) / I_final.
Now, let's plug in the given values:
M (mass of platform) = 143.1 kg
R_initial (initial distance from center) = 3.01 m
ω_initial (initial angular velocity) = 2.10 rad/s
R_final (final distance from center) = 2.89 m
First, we calculate I_initial = M * R_initial^2:
I_initial = 143.1 kg * (3.01 m)^2
Next, we calculate I_final = M * R_final^2:
I_final = 143.1 kg * (2.89 m)^2
Now, we can substitute these values into the equation for ω_final:
ω_final = (I_initial * ω_initial) / I_final
After substituting the calculated values, you can solve the equation to find the value of ω_final in rad/s.