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?

coefficient of static friction=

Wt. of stone = m*g

Fp = mg*sin30=Force parallel to incline.

Fn = mg*cos30 = Normal force = Force
perpendicular to incline.

Fs = u*Fn = Force of static friction.

Fp-Fs = m*a = m*o = 0
Fp = Fs
mg*sin30 = u*mg*cos30
Divide both sides by mg:
sin30 = u*cos30
u = sin30/cos30 = 0.577 = Coefficient of static friction.

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To determine the coefficient of static friction between the block and the truck bed, we need to analyze the forces acting on the block. Let's break it down step by step:

1. Draw a free-body diagram: Draw a diagram showing the block and all the forces acting on it. In this case, we have the weight of the block acting vertically downward (mg), the normal force perpendicular to the truck bed, and the force of static friction acting parallel to the truck bed. The angle between the bed and the horizontal is 30 degrees.

2. Resolve the forces: Resolve the weight (mg) into its horizontal and vertical components. The vertical component is mg * cos(30), and the horizontal component is mg * sin(30).

3. Apply Newton's second law: According to Newton's second law, the sum of all forces acting on an object equals its mass times its acceleration. In this case, the acceleration is zero since the block is just on the verge of sliding.

4. Determine the sum of horizontal forces: In equilibrium, the sum of horizontal forces is zero. Since the block is not accelerating horizontally, the horizontal component of the weight must be balanced by the force of static friction.

5. Calculate the force of static friction: The force of static friction can be calculated using the equation F = μN, where μ is the coefficient of static friction and N is the normal force. In this case, the normal force is equal to the vertical component of the weight.

Now, substitute the known values into the equation from step 5 and solve for the coefficient of static friction:

mg * sin(30) = μ * mg * cos(30)

Cancel out the mass:

sin(30) = μ * cos(30)

Divide both sides by cos(30):

tan(30) = μ

Since tan(30) = 0.577, the coefficient of static friction between the block and the truck bed is approximately 0.577.