10. A large stone block has been loaded in the bed of a dump truck. The bed is slowly raised until it makes an angle of 30 degrees with the horizontal; at this point, the block starts to slide downward. What is the coefficient of static friction between the block and the truck bed?

=

force down the slide=friction

mg*sinTheta=mg*mu*cosTheta

tan theta= mu

To find the coefficient of static friction between the block and the truck bed, we first need to understand the forces acting on the block.

When the block is not moving (i.e., before it starts sliding downward), the force of static friction opposes the component of the force of gravity parallel to the truck bed. This means that the force of static friction is equal in magnitude and opposite in direction to the component of the force of gravity pulling the block down the incline.

The component of the force of gravity parallel to the incline can be calculated by multiplying the mass of the block (m) by the acceleration due to gravity (g) by the sine of the angle (θ) between the incline and the horizontal plane. In this case, θ is 30 degrees.

The formula for this component of the force of gravity is F_parallel = m * g * sin(θ).

Now, the force of static friction can be determined using the equation Fs <= µs * Fn, where Fs is the force of static friction, µs is the coefficient of static friction, and Fn is the normal force.

The normal force (Fn) is the force exerted by the truck bed on the block perpendicular to the incline. The normal force is equal in magnitude and opposite in direction to the component of the force of gravity perpendicular to the incline.

The component of the force of gravity perpendicular to the incline can be calculated by multiplying the mass of the block (m) by the acceleration due to gravity (g) by the cosine of the angle (θ) between the incline and the horizontal plane. In this case, θ is 30 degrees.

The formula for this component of the force of gravity is F_perpendicular = m * g * cos(θ).

Since the incline is at an angle of 30 degrees, we can use trigonometric functions to calculate the values of sin(30) and cos(30):

sin(30) = 0.5
cos(30) = √3/2

Now, substituting the values, the formula for the normal force becomes:

Fn = m * g * cos(θ) = m * g * (√3/2)

Finally, substituting the value of the normal force and the component of the force of gravity parallel to the incline, the expression for the force of static friction becomes:

Fs = µs * Fn = µs * m * g * (√3/2)

To find the coefficient of static friction (µs), we need to rearrange the equation:

µs = Fs / (m * g * (√3/2))

Note: The value of g is approximately 9.81 m/s^2.

By substituting the known values (such as the mass of the block and the angle of the incline) into the equation, you can solve for the coefficient of static friction between the block and the truck bed.