Question

1.)

An 7.2-kg box that has a uniform density and is twice as tall as it is wide rests on the floor of a truck. What is the maximum coefficient of static friction between the box and floor so that the box will slide toward the rear of the truck rather than tip when the truck accelerates forward on a level road?
μ _s <

2.)

Repeat the problem if the truck accelerates up a hill that makes an angle of 8.6° with the horizontal.
μ _s <

Answer 1

1.

Answer 2

Problem 1: Box on Level Road :-

Given:

• Mass $m=7.2\text{kg}$

• Height $h=2w$ (twice as tall as wide)

Condition for Tipping:

• Torque balance about front edge:

  $$mg\left(\frac{w}{2}\right)=ma(h)⟹\frac{w}{2}=a(2w)⟹a=\frac{g}{4}$$

• Sliding condition:

  $$ma<{\mu }_{s}mg⟹{\mu }_s>\frac{a}{g}=\frac{1}{4}$$

• To prevent tipping (sliding first):

  $${\mu }_s<0.25$$

Answer:

$$\boxed{{\displaystyle {\mu }_s<0.25}}$$

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Problem 2: Box on $8.6^{\circ }$ Hill  :-

Given:

• Hill angle $\theta =8.6^{\circ }$

Condition for Tipping:

• Torque balance about front edge:

  $$mg\cos \theta \left(\frac{w}{2}\right)=ma(2w)+mg\sin \theta (2w)$$$$\frac{g\cos \theta }{4}=a+g\sin \theta ⟹a=\frac{g\cos \theta }{4}-g\sin \theta$$

• Sliding condition:

  $$ma+mg\sin \theta <{\mu }_{s}mg\cos \theta$$

  Substitute $a$:

  $$\frac{g\cos \theta }{4}-g\sin \theta +g\sin \theta <{\mu }_{s}g\cos \theta ⟹\frac{1}{4}<{\mu }_s$$

• To prevent tipping (sliding first):

  $${\mu }_s<0.25$$

Answer:

$$\boxed{{\displaystyle {\mu }_s<0.25}}$$

Answers:

1. Level road: $\boxed{{\displaystyle {\mu }_s<0.25}}$

2. $8.6^{\circ }$ hill: $\boxed{{\displaystyle {\mu }_s<0.25}}$