A crate of potatoes of mass 13.0 kg is on a ramp with angle of incline 30° to the horizontal. The coefficients of friction are μs = 0.76 and μk = 0.35. Find the frictional force (magnitude and direction) on the crate if the crate is sliding up the ramp.

the crate is sliding, so μk (the kinetic coefficient) is the relevant one

the direction of the frictional force is OPPOSITE to the direction of motion

the magnitude of the force is: the normal force of the crate on the ramp [m * g * cos(30º)]; multiplied by the coefficient of kinetic friction (μk)

To find the frictional force on the crate, we first need to analyze the forces acting on it.

1. We know that there is a gravitational force acting vertically downward on the crate, which can be calculated as the product of its mass and the acceleration due to gravity: F_gravity = m * g, where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s^2).

F_gravity = (13.0 kg) * (9.8 m/s^2) = 127.4 N (rounded to the nearest tenth)

2. Next, we need to find the normal force acting perpendicular to the ramp. Since the crate is on an incline, the normal force is not equal to the weight of the crate (F_gravity). It can be calculated using the angle of incline (θ) and the gravitational force.

Normal force = F_gravity * cos(θ)

θ = 30°

Normal force = (127.4 N) * cos(30°) = 110.4 N (rounded to the nearest tenth)

3. Now, let's calculate the frictional force.

When the crate is sliding up the ramp, the frictional force is the force opposing the motion, so it acts in the opposite direction of the crate's motion.

The maximum static frictional force (fs) can be calculated as the product of the coefficient of static friction (μs) and the normal force:

fs = μs * Normal force

μs = 0.76 (given)

fs = (0.76) * (110.4 N) = 84.0 N (rounded to the nearest tenth)

Since the crate is sliding up the ramp, the frictional force is less than the maximum static frictional force. It can be calculated as the product of the coefficient of kinetic friction (μk) and the normal force:

fk = μk * Normal force

μk = 0.35 (given)

fk = (0.35) * (110.4 N) = 38.6 N (rounded to the nearest tenth)

Therefore, the magnitude of the frictional force on the crate is approximately 38.6 N, and it acts in the opposite direction of the crate's motion (up the ramp).