A spring with stiffness k s and relaxed length L stands vertically on a table. Y

Question

A spring with stiffness ks and relaxed length L stands vertically on a table. You hold a mass M just barely touching the top of the spring.

(a) You very slowly let the mass down onto the spring a certain distance, and when you let go, the mass doesn't move. How much did the spring compress? (Enter the stretch of the spring, including the proper sign. Use the following as necessary: M, g, ks.) (this answer was incorrect when s=Mg/k was inputted and s=Mg/k_s)

s =

How much work did you do? (Use the following as necessary: M, g, ks.) (this answer was incorrect when the following was inputted w= -{M^2g^2}{2k}

W =

(b) Now you again hold the mass just barely touching the top of the spring, and then let go. What is the maximum compression of the spring? (Enter the stretch of the spring, including the proper sign. Use the following as necessary: M, g, ks.)

s =

State what approximations and simplifying assumptions you made.

(c) Next you push the mass down on the spring so that the spring is compressed an amount s, then let go, and the mass starts moving upward and goes quite high. When the mass is a height of 2L above the table, what is its speed? (Use the following as necessary: M, g, ks, s, and L.)

Explanation / Answer

a)

How much did the spring compress?
Fs = ks W=mg
ks = -mg
Change in Length = -Mg/ks

**How much work did you do?

Wyou = -(1/2) * (M^2 * g^2)/ks

b)

What is the maximum compression of the spring?
Delta U = Ws
U = (1/2)ks^2
W=mgs

mgs= (1/2)ks^s
max compression = -2 * (Mg/ks)

c)

When the mass is a height of 2L above the table, what is its speed?
sqrt((ks/M) * s^2 - 2g(L+s))

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