A 4.5 kg box slides down a 5.2-m -high frictionless hill, starting from rest, ac

Question

A 4.5 kg box slides down a 5.2-m -high frictionless hill, starting from rest, across a 2.2-m -wide horizontal surface, then hits a horizontal spring with spring constant 510 N/m. The other end of the spring is anchored against a wall. The ground under the spring is frictionless, but the 2.2-m-long horizontal surface is rough. The coefficient of kinetic friction of the box on this surface is 0.23. What is the speed of the box just before reaching the rough surface? Express your answer to two significant figures and include the appropriate units. What is the speed of the box just before hitting the spring? Express your answer to two significant figures and include the appropriate units. How far is the spring compressed? Express your answer to two significant figures and include the appropriate units. Including the first crossing, how many complete trips will the box make across the rough surface before coming to rest?

Explanation / Answer

Given,

m = 4.5 kg ; h = 5.2 m ; d = 2.2 m ; k = 510 N/m ; u = 0.23

A)from energy conservation, PE at the hilltop will be equal to the KE at the bottom so,

PE = KE

m g h = 1/2 mv^2

v = sqrt (2 gh) = sqrt(2 x 9.8 x 5.2) = 10.1 m/s

Hence,v = 10.1 m/s

b)The PE at top is

PE = m g h = 4.5 x 9.8 x 5.2 = 229.32 J

The work done against friction on the rough surface is:

W = Ff d = u m g d

W = 0.23 x 4.5 x 9.8 x 2.2 = 22.31 J

KE at bottom = PE - W = 229.32 - 22.31 = 207 J

1/2 m v'^2 = 207 J

v' = sqrt (2 x 207/m) = sqrt (2 x 207/4.5) = 9.6 m/s

Hence, v' = 9.6 m/s

c)for this

KE at bottom = PE of spring

207 = 1/2 k x^2

x = sqrt (2 x 207/k) = sqrt (2 x 207/510) = 0.9 m

Hence, x = 0.9 m = 90 cm

d)everytime the box looses, W = 22.31 J of energy on the rough .

Intially the box has, PE = 229.32 J

so the number of trips will be:

n = PE/W = 229.32/22.31 = 10.28 trips

Hence, n = 10.28 trips

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.