c. Find the time when the box stops for the first time. · OlympiadBench #1384

Physics · mechanics

Problem

c. Find the time

t_{0}

when the box stops for the first time. Given constants: ## Toffee Pudding A box of mass

m

is at rest on a horizontal floor. The coefficients of static and kinetic friction between the box and the floor are

\mu_{0}

and

\mu

(less than

\mu_{0}

), respectively. One end of a spring with spring constant

k

is attached to the right side of the box, and the spring is initially held at its relaxed length. The other end of the spring is pulled horizontally to the right with constant velocity

v_{0}

. As a result, the box will move in fits and starts. Assume the box does not tip over. Context question: a. Calculate the distance

s

that the spring is stretched beyond its rest length when the box is just about to start moving. Context answer:

s=\mu_{0} m g / k

Context question: b. Let the box start at

x=0

, and let

t=0

be the time the box first starts moving. Find the acceleration of the box in terms of

x, t, v_{0}, s

, and the other parameters, while the box is moving. Context answer:

a=\frac{k}{m}\left(s+v_{0} t-x\right)-\mu g

Extra Supplementary Reading Materials: The position of the box as a function of time

t

as defined in part (b) is

x(t)=\frac{v_{0}}{\omega}(\omega t-\sin \omega t)+(1-r) s(1-\cos \omega t),

where

\omega=\sqrt{k / m}

and

r=\mu / \mu_{0}

. This expression applies as long as the box is still moving, and you can use it in the parts below. Express all your answers in terms of

v_{0}, \omega, s

, and

r

Browse the problem library