To solve this problem, we can use the conservation of angular momentum. The initial angular momentum of the system is the sum of the angular momentum of the carousel and the angular momentum of the person. Since the person is initially not moving, their initial angular momentum is zero. The final angular momentum of the system is the sum of the angular momentum of the carousel and the person when both are moving together with the same angular velocity.
Let's denote the initial angular velocity of the carousel as ω1, the final angular velocity as ω2, the moment of inertia of the carousel as I, and the mass and radius of the carousel as m and R, respectively.
Initial angular momentum (L1) = Iω1 + (mass of person) * (angular momentum of person)
Since the person is initially not moving,
L1 = Iω1
Final angular momentum (L2) = Iω2 + (mass of person) * (angular momentum of person)
Since both the carousel and the person are moving with the same final angular velocity,
L2 = Iω2 + (mass of person) * (R^2 * ω2)
According to the conservation of angular momentum, the initial and final angular momenta should be equal:
L1 = L2
Iω1 = Iω2 + (mass of person) * (R^2 * ω2)
Now we can plug in the given values to find the final angular velocity ω2:
ω1 = 3.42 rad/s
I = 122 kg * m^2
mass of person = 46.3 kg
R = 1.70 m
Substituting these values into the equation, we get:
(122 kg * m^2) * (3.42 rad/s) = (122 kg * m^2) * ω2 + (46.3 kg) * ((1.70 m)^2) * ω2
Simplify and solve for ω2:
417.64 kg * m^2/s = (122 + 46.3 * 1.7^2) kg * m^2 * ω2
ω2 = 417.64 kg * m^2/s / (122 + 46.3 * 1.7^2) kg * m^2
ω2 ≈ 2.21 rad/s
The final angular speed of the carousel after the person climbs aboard is approximately 2.21 rad/s.
As seen from above, a playground carousel is rotating counterclockwise about its center on frictionless bearings. A person standing still on the ground grabs onto one of the bars on the carousel very close to its outer edge and climbs aboard. Thus, this person begins with an angular speed of zero and ends up with a nonzero angular speed, which means that he underwent a counterclockwise angular acceleration. The carousel has a radius of 1.70 m, an initial angular speed of 3.42 rad/s, and a moment of inertia of 122 kg · m2. The mass of the person is 46.3 kg. Find the final angular speed of the carousel after the person climbs aboard.
1 answer
1 answer