A puck of mass m = 1.10 kg slides in a circle of radius r = 24.0 cm on a frictionless table while attached to a hanging cylinder of mass M = 3.00 kg by a cord through a hole in the table. What speed (in m/s) keeps the cylinder at rest?
To determine the speed that keeps the cylinder at rest, we can use the conservation of angular momentum. When the puck moves in a circular path, the angular momentum of the system remains constant. The angular momentum, L, is given by the product of the moment of inertia, I, and the angular velocity, ω.
In this case, the moment of inertia for the puck, I_puck, is given by the equation I_puck = m * r^2, where m is the mass of the puck and r is the radius of the circular path.
The moment of inertia for the cylinder, I_cylinder, can be approximated as I_cylinder = (1/2) * M * R^2, where M is the mass of the cylinder and R is its radius.
Since the puck and the cylinder are connected by a cord, they share the same angular velocity. Therefore, ω_puck = ω_cylinder = ω.
By the conservation of angular momentum, the initial angular momentum of the system is equal to the final angular momentum. Initially, the cylinder is at rest, so its angular momentum is zero. Thus, the final angular momentum is also zero.
The initial angular momentum is given by L_initial = I_puck * ω_initial + I_cylinder * ω_initial.
To keep the cylinder at rest, the final angular velocity, ω_final, is zero. Hence, the initial angular momentum equation becomes 0 = I_puck * ω_initial + I_cylinder * ω_initial.
Substituting the expressions for the moments of inertia, we get 0 = (m * r^2 + (1/2) * M * R^2) * ω_initial.
Simplifying the equation, we have 0 = (1.10 kg * (24.0 cm)^2 + (1/2) * 3.00 kg * R^2) * ω_initial.
Now, we need to convert the radius from centimeters to meters. Since 1 meter equals 100 centimeters, R = 24.0 cm * (1 m / 100 cm) = 0.24 m.
Substituting the values, the equation becomes 0 = (1.10 kg * (0.24 m)^2 + (1/2) * 3.00 kg * (0.24 m)^2) * ω_initial.
Now, let's simplify and solve for ω_initial:
0 = (0.31104 kg * m^2 + 0.5184 kg * m^2) * ω_initial.
0 = 0.82944 kg * m^2 * ω_initial.
To keep the cylinder at rest, the angular velocity, ω_initial, must be zero. Therefore, to find the speed that keeps the cylinder at rest, we need to calculate the linear velocity, v, using the equation v = ω_initial * r.
Since ω_initial is zero, the linear velocity, v, is also zero.
Therefore, the speed (in m/s) that keeps the cylinder at rest is 0 m/s.