Consider tension force on rope. When ball/cart is not moving, T = mbg. If ball swings to an angle theta, O, then the y-component of T is mbg cosO. The cart is also pulling on the crane of the rope with a Fapp that is opposite to the x-component of the T (mbg sinO).
Newton's Second Law says to find the Fnet in both directions, so for very small angles Fnet = Fapp - mbg sinO. Total mass is m + mb, so a (of ball) is nearly equal to (Fapp - mbg sinO)/(m +mb). Once we find a, then we can use kinematics to solve for velocity and position at any given point in time.
A cart of mass m (including the mass of the crane) has a pendulum hanging from a crane attached to the cart. There is no friction due to the normal force, u = 0, or wheel bearings of the cart wheels. The cart is experiencing and applied force to the left of 10N. The ball at the end of massless rope attached to the crane has a mass mb and the length of the rope tied to the crane and ball is length, L. Express the motion of the ball in terms of m, mb,Fapp (=10N), L and g (acceleration of gravity near the surface of the earth).
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