## Pulley + Belt

In the figure shown, a pulley is driven by a belt. What minimum coefficient of friction will prevent slipping between pulley and belt? Ignore centrifugal effects.

##
__
__**Hint**

**Hint**

Belt Friction:

$$$F_1=F_2e^{\mu \theta}$$$

where
$$F_1$$
is the applied force in the direction of impending motion,
$$F_2$$
is the applied force resisting impending motion,
$$\mu$$
is the static coefficient of friction, and
$$\theta$$
is the total angle of contact between surfaces in radians.

##
__
__**Hint 2**

**Hint 2**

Unit Circle:

Belt Friction:

$$$F_1=F_2e^{\mu \theta}$$$

where
$$F_1$$
is the applied force in the direction of impending motion,
$$F_2$$
is the applied force resisting impending motion,
$$\mu$$
is the static coefficient of friction, and
$$\theta$$
is the total angle of contact between surfaces in radians. Recall a unit circle to determine
$$\theta$$
:

Therefore,

$$$4,250N=(4,000N)e^{\mu \pi/2}$$$

Solving for the coefficient of friction:

$$$ln(\frac{4,250}{4,000})=ln(e^{\mu \pi/2})=\mu \frac{\pi }{2}$$$

$$$\mu=\frac{2}{\pi}\cdot ln(\frac{4,250}{4,000})=\frac{2}{\pi} \cdot 0.06=0.04$$$

0.04