A 2.90 kg box that is several hundred meters above the surface of the earth is s

Question

A 2.90 kg box that is several hundred meters above the surface of the earth is suspended from the end of a short vertical rope of negligible mass. A time-dependent upward force is applied to the upper end of the rope, and this results in a tension in the rope of T(t)=(37.0N/s)t . The box is at rest at t =0. The only forces on the box are the tension in the rope and gravity.

What is the maximum distance that the box descends below its initial position?

At what value of t does the box return to its initial position?

Explanation / Answer

here,

from free body diagram

mg - T(t) = ma(t)

a(t) = (mg - T(t)/m

a(t) = g - (37/2.9) t

a(t) = 9.8 - 12.75 t

Velocity as a function of time

a(t) = dV/dt

dV = a(t) dt

V = int of dV = int of (a(t) dt

V = int of ((g - 12.75t)

v(t) = gt - 1275 t^2/2

part A:

V at t= 1s is V(1) = (9.8*1) - (12.75 * 1*1/2)

Vat t = 1 = 3.425 m/s^2

part B :

V at t = 3s,

V(3) = 9.8*3 - (12.75 * 3*3/2)

V(3) = -28 m/s^2

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at t = 0.015 secs it returns

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