A steel block of mass m_b = 2.0 kg slides on a horizontal surface with no fricti

Question

A steel block of mass m_b = 2.0 kg slides on a horizontal surface with no friction at a speed of v_b = 1.0 m/s. The steel block encounters an un-stretched spring and compresses it a distance Delta x = 0.5 m before coming to rest. A) What is the spring constant (k_0) of the spring? What is the duration of time the steel block is in contact with the spring before it fully compresses the spring? If the spring constant is changed to k_new = 0.5k_0, then what is the duration of time that the steel block will be in contact with the new spring before coming to rest expressed in terms of the original contact time found in part B)?

Explanation / Answer

energy conservation

K.E = P.E stored in the spring

1/2 m V^2 = 1/2 ko (x)^2

2(1)^2 = ko *(0,5)^2

ko = 8 N/m

Force exerted by the spring when it is compressed by a distance x = -ko*x

so, ma = - ko*x

Now
a = dv / dt
= dv / dt * dx / dx
= dv / dx * dx / dt
= v dv / dx

Integrate both sides (LHS within the limits u to v and RHS within the limits 0 to x; u is the initial velocity = 1 m/s)  

x^2 = m/ko (u^2 - v^2)

Also
a = dv / dt

So
- kox / m = dv / dt
Substitute the value of x from (1):
dv / dt = - sqrt(ko/m) * sqrt(u^2 - v^2)
or
dv = - sqrt(ko/m) * sqrt(u^2 - v^2) * dt

Integrate LHS within the limits u to v and RHS within the limits 0 to t :

Simplify to get
t = sqrt (m / ko) [ sin^-1 (1) - sin^-1 (v / u) ]

Put v (final velocity) = 0
t = sqrt (m / ko) [ sin^-1 (1) - sin^-1 (0) ]

Substitute sin^-1 (1) = pi/2 and sin^-1 (0) = 0  

Actually sin^-1 (x) can have multiple values but here you need to find the minimum value of the expression " sin^-1 (1) - sin^-1 (0) ")

t = sqrt (m / k) * pi / 2

= sqrt(2/8)*pi/2 = 0.78 s

t for knew = sqrt(2/4)*pi/2 = 1.11 s

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