A 4.10 kg mass, m, on a frictionless table is moving in a circle with radius 0.390 m at a constant speed. m is attached to a 5.00 kg mass, M, by a cord through a hole in the table. Find the speed with which m must move for M to stay at rest.
To solve this problem, we can use the concept of centripetal force and the principle of conservation of angular momentum. Here are the steps to find the speed with which m must move for M to stay at rest:
Step 1: Identify the forces acting on the masses.
In this problem, the only force acting on the masses is the tension in the cord. The tension provides the centripetal force necessary to keep the mass m moving in a circle.
Step 2: Set up the equations.
The centripetal force on mass m is given by F = m * (v^2 / r), where m is the mass of m, v is the velocity of m, and r is the radius of the circular motion.
Step 3: Apply the conservation of angular momentum.
Angular momentum is conserved when the net torque acting on a system is zero. In this case, the angular momentum of the system is given by L = m * v * r, where m is the mass of m, v is its velocity, and r is the radius of the motion. Since the angular momentum of the system is conserved, the initial angular momentum of mass m should be equal to the final angular momentum of masses m and M together. Mathematically, we can express this as m * v * r = (m + M) * V * R, where V is the velocity of M and R is the radius of the circular motion of M.
Step 4: Solve the equations simultaneously.
Substituting the expressions for centripetal force and angular momentum into the conservation equation, we get:
m * (v^2 / r) = (m + M) * V * R
Simplifying the equation gives:
v = (R * M * V) / (r * (m + M))
Step 5: Solve for v.
We need to find the speed with which m must move for M to stay at rest. This means V = 0, so the equation becomes:
v = (R * M * 0) / (r * (m + M))
Since V = 0, we have:
v = 0
Therefore, the speed with which m must move for M to stay at rest is zero.