Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 6 - Dynamics I: Motion Along a Line - Exercises and Problems - Page 164: 54

Answer

$23.1\;\rm N$

Work Step by Step

Since the box slides down, so the friction force must be up. See the force diagram of the box below. Since we need the box to move down at a constant velocity, the net force exerted on it must be zero. $$\sum F_x=F_n-F_{push}\cos45^\circ=ma_x=m(0)=0$$ Thus, $$F_n=F_{push}\cos45^\circ\tag 1$$ $$\sum F_y=f_k+F_{push}\sin45^\circ-mg=ma_y=m(0)=0$$ Hence, $$f_k+F_{push}\sin45^\circ-mg=0$$ $$\mu_kF_n+F_{push}\sin45^\circ =mg $$ Plugging from (1); $$\mu_kF_{push}\cos45^\circ+F_{push}\sin45^\circ=mg $$ $$F_{push}\left[\mu_k\cos45^\circ+ \sin45^\circ\right]=mg $$ $$F_{push}=\dfrac{mg}{\left[\mu_k\cos45^\circ+ \sin45^\circ\right]} =\dfrac{2\cdot 9.8}{\left[0.2\cos45^\circ+ \sin45^\circ\right]} $$ $$F_{push}=\color{magenta}{\bf23.1}\;\rm N$$
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