Question

A solid ball of mass M is made to fall from height H on pan suspended through a spring constant K as shown in figure. If the ball does not rebound and the pan is massless then amplitude of oscillation is.... ​

Answer 1

force due to gravity on the ball= mg= force exerted by ball on spring.

force exterted by spring to the ball mass m= -kX

hence by Newton's third law

mg= kX

Now

potential energy(P.E) of the ball to Amptlitude A is

= mg(H+X+A)

kinetic energy (K.E) of spring = -1/2{k(X+A)^2}

from law of conservation of energy

k.E+P.E= 0

hence

K.E= P.E

1/2{k(X+A)^2} = mg(H+X+A)

mgH +mgx + mgA= ½ kA^2 +kxA + ½ kx^2

we know mg =kX, replace kX by mg

mgH +mg(mg/k)+mgA = ½ kA^2 + mgA + ½ k(mg/k)^2

mgH +(mg)^2/k = ½kA^2 +½(mg)^2/k

½ kA^2 = mgH +½ (mg)^2/k

A = square root of [ 2mgH/k + (mg/k)^2]

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