## anonymous one year ago You push against a steamer trunk with a force of 800 N at an angle alpha with the horizontal . The trunk is on a flat floor and the coefficient of static friction between the trunk and floor is 0.55. The mass of the trunk is 87 kg. What is the largest value of alpha that will allow you to move the trunk?

1. Michele_Laino

the situation of your problem is described by the subsequent drawing: |dw:1438333970203:dw|

2. Michele_Laino

the pressure on the floor has the subsequent magnitude: $\Large mg + F\sin \theta$ whereas the driving force, has the subsequent magnitude: $\Large F\cos \theta$ the trunk will move, if and only if the subsequent condition is checked: $\Large F\cos \theta > \mu \left( {mg + F\sin \theta } \right)$ where \mu is the coefficient of static friction

3. Michele_Laino

hint: after a simplification, we can write: $\Large \cos \theta - \mu \sin \theta > \frac{{\mu mg}}{F}$ if we divide last condition by: $\Large \sqrt {{\mu ^2} + 1}$ we get: $\Large \frac{1}{{\sqrt {{\mu ^2} + 1} }}\cos \theta - \frac{\mu }{{\sqrt {{\mu ^2} + 1} }}\sin \theta > \frac{1}{{\sqrt {{\mu ^2} + 1} }}\frac{{\mu mg}}{F}$ which can be rewritten as follows: $\Large \cos \left( {\theta - \varphi } \right) > \frac{1}{{\sqrt {{\mu ^2} + 1} }}\frac{{\mu mg}}{F}$ where \phi is such that: $\Large \tan \varphi = \mu$

4. Michele_Laino

oops.. I have made a typo, here is the right formula: $\Large \cos \left( {\theta + \varphi } \right) > \frac{1}{{\sqrt {{\mu ^2} + 1} }}\frac{{\mu mg}}{F}$