a 15-kg box is resting on a hill forming an angle with the horizontal. The coefficient of static friction for the box on the surface is 0.45. calculate the maximum angle of the incline just before the box starts to move

mgsinα=μ mgcosα

tanα=μ
α=arctan0.45 =24.2°

To calculate the maximum angle of the incline just before the box starts to move, we need to consider the forces acting on the box.

The force of gravity acting on the box can be calculated by multiplying the mass of the box (15 kg) by the acceleration due to gravity (9.8 m/s^2):

Force of gravity = mass * acceleration due to gravity
Force of gravity = 15 kg * 9.8 m/s^2
Force of gravity = 147 N

The maximum force of static friction can be found by multiplying the normal force acting on the box by the coefficient of static friction. The normal force is the force perpendicular to the surface and can be calculated by multiplying the force of gravity by the cosine of the angle of the incline:

Normal force = Force of gravity * cos(angle)

Since the box is at rest, the maximum frictional force will equal the gravitational force, meaning the static frictional force will equal the maximum static frictional force.

Maximum static frictional force = Force of gravity * sin(angle)

Setting the maximum static frictional force equal to the force of static friction (maximum frictional force before the box starts moving), we can solve for the angle:

Maximum static frictional force = Force of static friction
Force of gravity * sin(angle) = Force of gravity * coefficient of static friction
sin(angle) = coefficient of static friction
angle = arcsin(coefficient of static friction)

Plugging in the given coefficient of static friction (0.45) into the equation, we can now solve for the maximum angle:

angle = arcsin(0.45)
angle ≈ 26.6 degrees

Therefore, the maximum angle of the incline just before the box starts to move is approximately 26.6 degrees.

To calculate the maximum angle of the incline just before the box starts to move, we need to consider the forces acting on the box. There are two main forces to consider:

1. The force pulling the box down the incline is the force due to gravity, which is the weight of the box. The weight can be calculated as the mass of the box multiplied by the acceleration due to gravity. Thus, the weight of the box is (15 kg) x (9.8 m/s^2) = 147 N.

2. The force opposing the motion of the box down the incline is the static friction force. The maximum static friction force can be calculated as the coefficient of static friction multiplied by the normal force. The normal force is equal to the weight of the box multiplied by the cosine of the angle of the incline (since the incline is not vertical). Thus, the maximum static friction force is (0.45) x (147 N) x (cosθ), where θ is the angle of the incline.

To determine the maximum value of θ just before the box starts to move, the static friction force must equal its maximum value. Therefore, we can write the equation as follows:

(0.45) x (147 N) x (cosθ) = (15 kg) x (9.8 m/s^2)

Now we can solve for θ:

cosθ = (15 kg x 9.8 m/s^2) / (0.45 x 147 N)
cosθ = 1.372
θ = arccos(1.372)

Using a calculator, the arccos(1.372) is not a valid value since the range of cosθ is -1 to 1. Therefore, there is no angle θ that satisfies the equation, meaning the box will not start to move on this incline.

In conclusion, the maximum angle of the incline just before the box starts to move is undefined in this scenario.