A small stone of mass m is stuck on the periphery of a wheel of radius R rolling along the x-axis with a constant speed v. The centre of the wheel was at x = 0 at time t = 0, and stone was on the ground at that time. Find the angular momentum of the stone and its rate of change as a function of time. Where does the torque required for the change of angular momentum come from?
To find the angular momentum of the stone and its rate of change as a function of time, we need to understand the motion of the stone and the wheel.
The angular momentum of an object can be calculated using the formula: L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
In this case, the stone is stuck on the periphery of the wheel, so its motion is a combination of rotational and translational motion.
The moment of inertia, I, for a point particle of mass m located at a distance r from an axis of rotation is given by: I = m * r^2.
Since the stone is stuck on the periphery of the wheel, its distance from the axis of rotation is equal to the radius of the wheel, R.
Therefore, the moment of inertia for the stone is: I = m * R^2.
The stone's motion can be described by its linear velocity, v, along the x-axis. The angular velocity, ω, is related to the linear velocity by the formula: ω = v / R.
Now we can calculate the angular momentum of the stone: L = I * ω = (m * R^2) * (v / R) = m * R * v.
The angular momentum of the stone is given by: L = m * R * v.
The rate of change of angular momentum, dL/dt, can be found by taking the derivative of L with respect to time: dL/dt = d(m * R * v)/dt.
Since the mass of the stone, m, and the radius of the wheel, R, are constant, we can treat them as constants when differentiating.
Therefore, dL/dt = R * d(m*v)/dt.
Now, let's consider where the torque required for the change of angular momentum comes from.
Torque, τ, is the rotational analog of force and is given by the formula: τ = I * α, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
We can rewrite the torque formula as τ = dL/dt, since torque is equal to the rate of change of angular momentum.
Therefore, the torque required for the change of angular momentum comes from the forces acting on the stone-wheel system. These forces apply a torque to the system, causing the angular momentum to change.
In this case, the only force acting on the system is the friction between the wheel and the ground, which provides the torque necessary for the change of angular momentum.
In summary, the angular momentum of the stone is given by L = m * R * v, and the rate of change of angular momentum is dL/dt = R * d(m*v)/dt. The torque required for the change of angular momentum comes from the friction between the wheel and the ground.