Chapter 5, Problem 86A

a.

To determine

To discuss:Factor on which angle of tilting depends.

Explanation of Solution

Introduction:

Friction:

It is a contact force which acts between two surfaces. When two surfaces try to slide or slides across each other, friction force generates. Force of friction is always in opposite to direction in which any object is moving or trying to move. Friction opposes motion.

The Free body diagram is shown below,

  

Here, with increase in tilt angle, the sine component of the weight $F_g\sin \theta$ increases which is balanced by the static friction $F_f$ .

But after a certain angle $F_g\sin \theta >F_f$ and person will start to slide.

Hence,

The angle of tilting depends upon the static friction between the patient and the bedsheet.

After a certain maximum angle, the friction can’t hold the patient and the patient starts to slide.

b.

To determine

Coefficient of static friction.

Answer to Problem 86A

Coefficient of static friction between the patient and the bedsheet is ${\mu }_S=1$

Explanation of Solution

Given:

Maximum angle of tilt, $\theta =45°$

Formula used:

Coefficient of static friction is the ratio of frictional force to normal force.

Calculation:

The free body diagram can be drawn as,

  

At maximum angle, the component of weight parallel to bed will be equal to friction force that is

  $\begin{array}{c}{F}_g=F_f\\ mg\sin \theta ={\mu }_S{F}_N\left(\because F_f={\mu }_S{F}_N\right)\\ mg\sin \theta ={\mu }_{S}mg\cos \theta \left(\because F_N=mg\cos \theta \right)\\ {\mu }_S=\frac{mg\sin \theta }{mg\cos \theta }\\ =\frac{\sin \theta }{\cos \theta }\\ =\tan \theta \\ =\tan 45°\\ {\mu }_S=1\end{array}$

Conclusion:

Coefficient of static friction between the patient and the bedsheet is ${\mu }_S=1$ .