a mass is at rest with respect to the lab frame, while a frictionless turntable rotates beneath it. the frequency of turntable is w and mass is located at radius r. In the frame of turntable , find the forces acting on mass and verify that F=ma
To find the forces acting on the mass in the frame of the turntable, let's break it down step by step:
1. In the frame of the turntable, the mass appears to be moving in a circular path due to the rotation of the turntable beneath it. The apparent centrifugal force experienced by the mass is given by F_c = mω²r, where m is the mass of the object, ω is the angular velocity (frequency) of the turntable, and r is the radius from the center of rotation.
2. Since the turntable is frictionless, there is no friction force acting on the mass in the frame of the turntable.
3. Additionally, there is a normal force acting on the mass to counterbalance the apparent centrifuigal force. The normal force, N, is the force exerted by the surface of the turntable on the mass to prevent it from falling through.
4. In the frame of the turntable, where the mass appears to be in equilibrium, the net force acting on the mass is zero. Therefore, F_net = F_c + N = 0.
Now, let's verify the equation F = ma, where F is the net force acting on the mass and a is the acceleration of the mass.
1. The mass is at rest with respect to the lab frame, so its acceleration is zero, a = 0.
2. From the previous step, we found that F_net = F_c + N = 0. Since F_net = ma, we can substitute a = 0, giving us F_c + N = 0.
3. Rearranging the equation, we get F_c = -N.
4. We know that F_c = mω²r from step 1. Substituting this into the equation, we have mω²r = -N.
5. Both m and r are positive quantities. Therefore, for the equation to hold true, N must be negative to cancel out the positive term on the right side.
Hence, the verifying equation F = ma holds true since the net force acting on the mass is zero in the frame of the turntable when considering the apparent centrifugal force and the normal force.