A block with mass m = 16 kg rests on a frictionless table and is accelerated by

Question

A block with mass m = 16 kg rests on a frictionless table and is accelerated by a spring with spring constant k = 5052 N/m after being compressed a distance x1 = 0.556 m from the spring’s unstretched length. The floor is frictionless except for a rough patch a distance d = 2.7 m long. For this rough path, the coefficient of friction is k = 0.48.

1) How much work is done by the spring as it accelerates the block?

2) What is the speed of the block right after it leaves the spring?

3) How much work is done by friction as the block crosses the rough spot?

4) What is the speed of the block after it passes the rough spot?

5) Instead, the spring is only compressed a distance x2 = 0.159 m before being released. How far into the rough path does the block slide before coming to rest?

6) What distance does the spring need to be compressed so that the block will just barely make it past the rough patch when released?

7) If the spring was compressed three times farther and then the block is released, the work done on the block by the spring as it accelerates the block is:

a) the same

b) three times greater

c) three times less

d) nine times greater

e) nine times less

Explanation / Answer

Here,

m = 16 Kg

k = 5052 N/m

x1 = 0.556 m

d = 2.7 m

uk = 0.48

1)

work done by spring while it accelerate the block = 0.5 K * x^2

work done by spring while it accelerate the block = 0.5 * 5052 * 0.556^2

work done by spring while it accelerate the block = 781 J

2)

let the speed of block is u

Using Work energy theorum

work done by spring while it accelerate the block = 0.5 * m * u^2

781 = 0.5* 16 * u^2

u = 9.88 m/s

the speed of block just after leaving the spring is 9.88 m/s

3)

Work dony by friction during rough patch = - m * g * d * uk

Work dony by friction during rough patch = -16 * 9.8 * 2.7 * 0.48

Work dony by friction during rough patch = - 203.2 J

the Work dony by friction during rough patch is -203.2 J

4)

let the final speed of block is v

Using work energy theorum

0.5 * m * v^2 = 781 - 203.2

0.5 * 16 * v^2 = 781 - 203.2

v = 8.5 m/s

the speed of block after it passes the rough patch is 8.5 m/s

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