A pendulum consists of a compact 0.10 kg bob attached to a string of length 2.30

Question

A pendulum consists of a compact 0.10 kg bob attached to a string of length 2.30 m. A block of mass m rests on a horizontal frictionless surface. The pendulum is released from rest at an angle of 53° with the vertical. The bob collides elastically with the block at the lowest point in its arc. Following the collision, the maximum angle of the pendulum with the vertical is 5.73°. Determine the mass m (there are two possible values)
Larger value : ____ kg
Smaller value: ____ kg

Can a explanation be included as well I'm having problems with these types of problems.

Explanation / Answer

mass of the bob M = 0.10 kg, length of the string L = 2.3 m angle o = 53° and maximum angle = 5.73°. let the velocity of M before the collision = v from law of conservation of energy ,
         (1/2) Mv2 = MgL[1 - cos0]    .................... (1) velocity v = [2gL(1 - cos0]1/2   ................ (2) let the velocity of mass M after the elastic collision v' .             v' = (M - m)(v)/(M + m)            v'/v = (M - m) / (M + m) .................... (3) after collision , from law of conservation of energy,           (1/2)Mv'2 = MgL[1 - cos]   ................. (4) dividing the eq (4) with eq (1), we get        (v'/v)2 = [1 - cos]/[1 - cos0]                  = [1 - cos(5.73)]/[1 - cos(53)]                  = 0.0125            v'/v = ± 0.112 from eq (3) , we have (M - m)/(M + m) = v'/v if  v'/v = 0.112 , then (0.1 kg - m)/(0.1 kg + m) = 0.112 (0.1 kg - m) = (0.112)(0.1 kg + m) therefore , mass m = 0.079 kg from eq (3) , we have (M - m)/(M + m) = v'/v if  v'/v = 0.112 , then (0.1 kg - m)/(0.1 kg + m) = 0.112 (0.1 kg - m) = (0.112)(0.1 kg + m) therefore , mass m = 0.079 kg (0.1 kg - m) = (0.112)(0.1 kg + m) therefore , mass m = 0.079 kg if  v'/v = -0.112 , then (0.1 kg - m)/(0.1 kg + m) = -0.112 (0.1 kg - m) = -(0.112)(0.1 kg + m) therefore , mass m = 0.24 kg if  v'/v = -0.112 , then (0.1 kg - m)/(0.1 kg + m) = -0.112 (0.1 kg - m) = -(0.112)(0.1 kg + m) therefore , mass m = 0.24 kg

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