A 10-kg crate on a ramp is inclined at 30 degrees from the horizontal and released. The coefficients of friction between them are .48 (coefficient of static friction) and .28 (coefficient of kinetic friction).

a) Does the block slide?
b) What then is the actual friction force?

I'm a little confused on this so any help would be greatly appreciated.

component of gravity force down ramp =10*9.8*sin 30

= 49 N

component of gravity force normal to ramp = 10*9.8*cos 30
=84.9 N

Max static friction force = .48*84.9 = 40.7 N
so
gravity wins, it slides and friction coef becomes .28

friction force then = .28*84.9 = 23.8 N

To determine whether the block will slide or not, we need to compare the force component along the ramp due to gravity to the maximum friction force.

Let's start by calculating the force component along the ramp due to gravity. We can use the formula:

Force along the ramp = m * g * sin(θ)

Where:
m = mass of the crate = 10 kg
g = acceleration due to gravity = 9.8 m/s^2
θ = angle of inclination = 30 degrees

Plugging in these values, we get:
Force along the ramp = 10 kg * 9.8 m/s^2 * sin(30 degrees)
= 10 kg * 9.8 m/s^2 * 0.5
= 49 N

Now, let's determine the maximum static friction force that can be exerted on the crate. We can calculate it using the formula:

Maximum static friction force = coefficient of static friction * normal force

The normal force is the force acting perpendicular to the surface of the ramp, which is equal to the weight of the crate.

Normal force = m * g * cos(θ)
= 10 kg * 9.8 m/s^2 * cos(30 degrees)
= 10 kg * 9.8 m/s^2 * (√3 / 2)
≈ 84.55 N

Maximum static friction force = 0.48 * 84.55 N
≈ 40.58 N

Since the force along the ramp (49 N) is greater than the maximum static friction force (40.58 N), the block will slide down the ramp. The static friction force is not enough to prevent the crate from sliding.

Now, to calculate the actual friction force when the block is sliding, we can use the formula:

Friction force = coefficient of kinetic friction * normal force

The normal force remains the same as before (84.55 N).

Actual friction force = 0.28 * 84.55 N
≈ 23.62 N

Therefore, the actual friction force acting on the block while sliding down the ramp is approximately 23.62 N.