Page 85 - Schaum's Outline of Theory and Problems of Applied Physics

70 FRICTION [CHAP. 6

F w
u
N
u
w
u
Fig. 6-2

F w = w sin θ N = w cos θ

When the block just begins to move, the downward force along the plane F w must be equal to the maximum force
µ s N of static friction, so
F w = µ s N
w sin θ = µ s w cos θ
sin θ
µ s = = tan θ
cos θ
Thus the coefficient of static friction equals the tangent of the angle of repose. The weight of the block has no
significance here.

SOLVED PROBLEM 6.7

A lathe mounted on wooden skids is to be slid down a pair of planks placed against the back of a truck.
(a) If the coefficient of friction is 0.28, what angle should the planks make with the ground in order for
the lathe to slide down at constant velocity? (b) When the planks are at this angle, will the lathe start to
slide down of its own accord?
(a) When the lathe moves at constant velocity, there is no net force on it, according to Newton’s first law of motion.
The downward force due to the lathe’s weight therefore exactly balances the retarding force of sliding friction,
and by the reasoning of Prob. 6.6,

tan θ = µ k = 0.28 θ = 16 ◦
(b) Since the coefficient of static friction exceeds that of sliding friction, the lathe will have to be given a push to
start it moving.

SOLVED PROBLEM 6.8
A woman whose shoes have leather soles and heels is able to stand without slipping on a wooden surface
◦
that makes an angle of 25 with the horizontal. What is the minimum coefficient of static friction for
leather on wood?
Since the angle of repose is at least 25 ,
◦
◦
µ s ≥ tan 25 ≥ 0.47
where the symbol ≥ means “greater than or equal to.”

SOLVED PROBLEM 6.9
◦
A 5000-ton ship rests on launching ways that slope down to the water at an angle of 10 . If the coefficient
of sliding friction is 0.18, how much force is needed to winch the ship down the ways into the water?