A uniform ladder weighing 197 N is leaning against a wall. The ladder slips when θ = 63.0°. Assuming that the coefficients of static friction at the wall and the ground are the same, obtain a value for µs.

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To find the value of µs (the coefficient of static friction), we need to use the given information about the ladder's weight and the angle at which it slips.

Let's break down the problem and understand its components:

1. The weight of the ladder (W): The problem states that the ladder weighs 197 N.

2. The angle at which the ladder slips (θ): The problem states that the ladder slips at an angle of θ = 63.0°.

3. The forces acting on the ladder:

a. Normal force (N): The normal force acts perpendicular to the surface of contact and is equal to the weight of the ladder or 197 N.

b. Friction force (f): The friction force acts parallel to the surface of contact and opposes the slipping motion of the ladder.

c. Force of static friction (fs): The force of static friction opposes the tendency of relative motion between two surfaces in contact when they are not slipping. This force keeps the ladder in equilibrium until it starts to slip.

4. The coefficient of static friction (µs): This is the value we need to determine.

To find the coefficient of static friction, we need to consider the equilibrium of forces acting on the ladder when it is about to slip:

1. In the vertical direction (y-axis), the forces are balanced. The normal force (N) cancels out the vertical component of the weight (Wsinθ):

N - Wsinθ = 0 (Equation 1)

2. In the horizontal direction (x-axis), the friction force balances the horizontal component of the weight (Wcosθ). However, the maximum friction force can be calculated using the equation:

fs ≤ µsN (Equation 2)

Substituting the value of N from Equation 1, we get:

fs ≤ µs(Wsinθ) (Equation 3)

Since the ladder is just about to slip, the maximum friction force fs equals the available static friction force, which is µsN. Therefore, we can rewrite Equation 3 as:

µsN = µs(Wsinθ) (Equation 4)

Now we can substitute the known values into Equation 4 and solve for µs:

µs(197 N) = µs(197 N)(sin63.0°)

Cancelling out the common factor of 197 N:

µs = sin63.0°

Evaluating the sine of the angle:

µs ≈ 0.891

Therefore, the value of µs (the coefficient of static friction) is approximately 0.891.