# Chapter 5 - Exercises and Problems - Page 88: 56

The proof is below.

#### Work Step by Step

We know that you are trying to push the trunk forward, and the coefficient of friction is resisting this motion. Thus, we know that at any coefficient of friction at or beyond the coefficient of friction where the force of the push and the force of friction cancel will result in the box being impossible to move. Thus, we set the forces equal: $F_f = Fcos\theta\\ F_n \mu = Fcos\theta \\ (Fsin\theta + mg)(\mu)= Fcos\theta$ We know that if you cannot move the trunk, that means that the force due to your push will increase the normal force by more than your push will cause the trunk to move forward. Thus, it follows: $(Fsin\theta )(\mu)= Fcos\theta$ $\mu=\frac{cos50}{sin50}=.84$

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