A 2.00-kg sphere is rotating about an axis through its center at 40.0 rev/s with the angular velocity in the +z direction. A torque 10.0 Nm acts on the sphere about the center of the sphere in the +x direction. What is the rate of change of the angular momentum of the sphere?
A. -20.0 kg m2/s2.
B. 20.0 kg m2/s2.
C. 80.0 kg m2/s2.
D. 8.00 kg m2/s2.
E. 10.0 kg m2/s2.
Well, it seems like the angular momentum is going to have a change of heart. But don't worry, I'll calculate it for you.
The rate of change of angular momentum is given by the formula:
Rate of change of angular momentum = Torque
Since the torque acting on the sphere is 10.0 Nm, the rate of change of angular momentum will also be 10.0 kg m^2/s^2.
So, the correct answer is E. 10.0 kg m^2/s^2.
Looks like the angular momentum is keeping its momentum after all!
First, we need to calculate the initial angular momentum of the sphere. The angular momentum (L) is given by the equation L = Iω, where I is the moment of inertia and ω is the angular velocity.
The moment of inertia for a solid sphere rotating about an axis through its center is given by the equation I = (2/5)mr^2, where m is the mass of the sphere and r is the radius of the sphere.
Given that the mass of the sphere is 2.00 kg and the radius is not provided, we cannot calculate the moment of inertia directly. However, since we are interested in the rate of change of the angular momentum and not the actual angular momentum, we can ignore the constant factors and focus on the variables involved.
Therefore, the initial angular momentum (L_initial) can be approximated as L_initial ≈ mr^2ω.
Next, we need to calculate the final angular momentum (L_final) after the torque is applied. The torque (τ) is given by the equation τ = dL/dt, where dL/dt is the rate of change of the angular momentum.
Rearranging the equation, we have dL = τ dt. Integrating both sides yields ΔL = τ Δt, where ΔL is the change in angular momentum and Δt is the change in time.
Given that the torque acting on the sphere is 10.0 Nm and the time is not provided, we cannot calculate the change in angular momentum directly. However, since we are interested in the rate of change of the angular momentum and not the actual change in angular momentum, we can ignore the constant factors and focus on the variables involved.
Therefore, the rate of change of the angular momentum (dL/dt) can be approximated as dL/dt ≈ τ.
Substituting the given values, we have dL/dt ≈ 10.0 Nm.
Therefore, the rate of change of the angular momentum of the sphere is approximately 10.0 kg m^2/s^2.
Choice E, 10.0 kg m^2/s^2, is the correct answer.
To find the rate of change of the angular momentum of the sphere, you can use the equation:
Torque = Rate of Change of Angular Momentum
The torque acting on the sphere is given as 10.0 Nm in the +x direction.
The angular momentum of a rotating object is given by the equation:
Angular Momentum = (moment of inertia) * (angular velocity)
The moment of inertia of a sphere rotating about an axis through its center is given by the equation:
Moment of Inertia = (2/5) * (mass) * (radius^2)
Since the sphere is rotating about an axis through its center, we can assume the radius of the sphere is 0. Therefore, the moment of inertia of the sphere is:
Moment of Inertia = (2/5) * (mass) * (0^2)
Since the radius is 0, the moment of inertia is also 0.
Substituting the values into the equation Torque = Rate of Change of Angular Momentum:
10.0 Nm = (0 kg m^2/s) * (angular velocity)
The rate of change of the angular momentum is 0 kg m^2/s.
Therefore, the answer is none of the given options.