Question 5.11: Experimental Determination of µs and µk The following is a s...
Experimental Determination of \pmb{\mu_s} and \pmb{\mu_k}
The following is a simple method of measuring coefficients of friction. Suppose a block is placed on a rough surface inclined relative to the horizontal as shown in Figure 5.19. The incline angle is increased until the block starts to move. Show that you can obtain \mu_s by measuring the critical angle \theta_c at which this slipping just occurs.
Conceptualize Consider Figure 5.19 and imagine that the block tends to slide down the incline due to the gravitational force. To simulate the situation, place a coin on this book’s cover and tilt the book until the coin begins to slide. Notice how this example differs from Example 5.6. When there is no friction on an incline, any angle of the incline will cause a stationary object to begin moving. When there is friction, however, there is no movement of the object for angles less than the critical angle.
Categorize The block is subject to various forces. Because we are raising the plane to the angle at which the block is just ready to begin to move but is not moving, we categorize the block as a particle in equilibrium.
Analyze The diagram in Figure 5.19 shows the forces on the block: the gravitational force m \overrightarrow{g}, the normal force \overrightarrow{n}, and the force of static friction \overrightarrow{f}_s. We choose x to be parallel to the plane and y perpendicular to it.
From the particle in equilibrium model, apply Equation 5.8 to the block in both the x and y directions:
\Sigma \overrightarrow{F}=0 (5.8)
(1) \sum F_x=m g \sin \theta-f_s=0
(2) \sum F_y=n-m g \cos \theta=0
Substitute mg = n/cos θ from Equation (2) into Equation (1):
(3) f_s=m g \sin \theta=\left(\frac{n}{\cos \theta}\right) \sin \theta=n \tan \theta
When the incline angle is increased until the block is on the verge of slipping, the force of static friction has reached its maximum value \mu_s n. The angle θ in this situation is the critical angle \theta_c. Make these substitutions in Equation (3):
\begin{aligned} \mu_s n=n \tan \theta_c\\ \mu_s = \tan \theta_c \end{aligned}We have shown, as requested, that the coefficient of static friction is related only to the critical angle. For example, if the block just slips at \theta_c=20.0^{\circ}, we find that \mu_s=\tan 20.0^{\circ}=0.364.
Finalize Once the block starts to move at \theta \geq \theta_c, it accelerates down the incline and the force of friction is f_k=\mu_k n.