## Physics (10th Edition)

Published by Wiley

# Chapter 4 - Forces and Newton's Laws of Motion - Problems - Page 120: 120

#### Answer

The maximum acceleration the truck can attain is $1.68m/s^2$

#### Work Step by Step

As the truck moves uphill with acceleration $a$, the crate also moves with acceleration $a$, meaning there has to be a net force exerted on it horizontally. As the figure illustrates, the horizontal forces are $f_s$ and $mg\sin\theta$, with $f_s$ being the force moving the crate uphill. According to Newton's 2nd Law, $$\sum F=f_s-mg\sin\theta=ma$$ But $f_s$ is limited to a certain value. If $a$ is high enough to surpass $f_s^{max}$, the crate will slip backwards. Therefore, $$f_s^{max}-mg\sin\theta=ma_{max}$$ We have $f_s^{max}=\mu_sF_N=\mu_smg\cos\theta$ So, $$\mu_smg\cos\theta-mg\sin\theta=ma_{max}$$ $$a_{max}=\frac{\mu_smg\cos\theta-mg\sin\theta}{m}=\mu_sg\cos\theta-g\sin\theta$$ We have $\mu_s=0.35, g=9.8m/s^2, \theta=10^o$. Therefore, $$a_{max}=1.68m/s^2$$

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