A stick of length l = 65.0 cm rests against a wall. The coefficient of static friction between stick and the wall and between the stick and the floor are equal. The stick will slip off the wall if placed at an angle greater than θ = 39.0 degrees. What is the coefficient of static friction, μs, between the stick and the wall and floor?
μs=
unanswered
tan (theta)= (2*u_s)/(1-u_s^2)
,solve for u_s
To find the coefficient of static friction, we can analyze the forces acting on the stick.
Let's consider the forces acting on the stick when it's on the verge of slipping:
1. The weight of the stick, acting downward (mg).
2. The normal force exerted by the floor on the stick, acting perpendicular to the floor's surface (N).
3. The frictional force between the stick and the floor, opposing the tendency to slip (fs).
4. The frictional force between the stick and the wall, opposing the tendency to slip (fw).
Since the coefficient of static friction is the same for both surfaces (stick-wall and stick-floor), we can write:
fs = μs * N (1)
fw = μs * N (2)
The stick will slip off the wall when the angle between the stick and the wall (�) exceeds the critical angle, which is given as θ = 39.0 degrees.
To express θ in terms of �, we can use trigonometry. Recall that the tangent of an angle is equal to the opposite side divided by the adjacent side:
tan(�) = fw/mg
tan(θ) = fw/mg (since θ and � are equal)
Rearranging the equation gives:
fw = mg * tan(θ) (3)
Substituting equation (2) into equation (3), we have:
μs * N = mg * tan(θ)
N = mg / μs * tan(θ) (4)
Substituting equation (4) into equation (1), we get:
fs = μs * mg / μs * tan(θ)
fs = mg / tan(θ) (5)
Now, let's consider the torque acting on the stick due to the weight. The torque produced by the weight must be balanced by the torque produced by the friction force on the wall. The torque equation is given as:
τ = fw * l * sin(θ)
But since fw = fs, we have:
τ = fs * l * sin(θ)
Substituting equation (5) into the torque equation:
τ = (mg / tan(θ)) * l * sin(θ)
Simplifying, we get:
τ = mg * l
Therefore, we can write:
mg * l = mg * l
Now, we can determine the coefficient of static friction (μs) by canceling the mass (m) from both sides of the equation:
l = l / tan(θ)
tan(θ) = l / l
Now, we can calculate the value of tan(θ):
tan(θ) = 65.0 cm / 65.0 cm
tan(θ) = 1
Therefore, the coefficient of static friction (μs) is 1.
μs = 1