Inclination angle of 38.0 degrees with respect to the horizontal. Suppose a truck on a street, for the truck to remain at rest, a static friction must exist. If an additional force with a magnitude equal to one-fourth the weight of the truck is applied uphill in order to hold the truck in place, what is the coefficient of static-friction?

To find the coefficient of static friction, we need to consider the forces acting on the truck and use the fact that the truck is at rest.

Let's break down the forces acting on the truck:

1. Weight (W): The force of gravity pulling the truck downward. Its magnitude is given as W.

2. Normal Force (N): The perpendicular force exerted by the road on the truck. Since the truck is on an incline, the normal force is not equal to its weight. Instead, it can be calculated as N = W * cos(θ), where θ is the inclination angle.

3. Static Friction (Ff): The force opposing the motion of the truck. It acts parallel to the incline and prevents the truck from sliding downhill. The magnitude of static friction is Ff = μs * N, where μs is the coefficient of static friction.

4. Additional Force (F): The force applied uphill to hold the truck in place. Its magnitude is given as one-fourth of the weight of the truck, so F = 1/4 * W.

Since the truck is at rest, the net force acting on it must be zero. We can sum up the forces in the vertical (y) direction and the forces in the horizontal (x) direction separately:

In the y-direction:
N - W * cos(θ) = 0 (Equation 1)

In the x-direction:
F - W * sin(θ) - Ff = 0 (Equation 2)

Now, let's solve these equations to find the coefficient of static friction:

From Equation 1, we can rearrange it to get:
N = W * cos(θ)

Substituting this value of N in Equation 2, we get:
F - W * sin(θ) - μs * N = 0

Plugging in the given values:
1/4 * W - W * sin(θ) - μs * W * cos(θ) = 0

Now, we can rearrange and solve for the coefficient of static friction (μs):

μs * W * cos(θ) = 1/4 * W - W * sin(θ)

μs = (1/4 - sin(θ)) / cos(θ)

By substituting the given angle of 38.0 degrees (convert to radians), we can calculate the coefficient of static friction.