Up and down an incline with friction: time, accelerations, sliding distance, friction coefficient.

00:00 Overview: in this problem, a block slides up and down an incline with friction. This is a big five-part question, and we are going to solve for the acceleration on the way up, the maximum sliding distance up the ramp, the maximum static friction coefficient for which the block breaks loose and turns around, the acceleration on the way down and finally the total time for the block to slide up the ramp and back to its original position. 🧠 Access full flipped physics courses with video lectures and examples at https://www.zakslabphysics.com/ 00:51 Acceleration on the way up the ramp: we diagram all forces on the block. We put the force of gravity into the diagram and break into perpendicular and parallel components, we quickly find the normal force, and we find the friction force on the way up the ramp. Applying Newton's second law a=F_net/m, we find the acceleration as the block slides up the ramp. 03:38 Max sliding distance up the ramp: now that we have the acceleration of the block on the way up the ramp, we can compute the total slide distance by using the time-independent kinematics equation v^2=v_0^2+2a(x-x_0). The key to the problem is that the final velocity is zero, so we plug everything in and solve for the final position of the block when it slides to a stop. 05:08 Find the maximum static friction coefficient for the block to break loose and turn around on the ramp. In this part, we diagram all the forces on the block after it stops, with static friction pointing up the ramp to hold the block in place. We simply assume the static friction force is maxed out at mu_s*n and that the forces are balanced on the block. This gives a static friction force that is barely holding on to the block, so it's an upper bound for the static friction coefficient (even an infinitesimally smaller coefficient will cause the block to break loose). We set mgsin(theta)=mu_s*mgcos(theta) and solve for the static friction coefficient. 07:23 Acceleration down the incline: now that the block is sliding down the ramp, the kinetic friction force reverses direction and points up the ramp. We make a new force diagram for the block and use Newton's second law once again to find the acceleration on the way down the ramp. 08:37 Total time for the block to slide up and down the incline: we compute the time for the block to slide up the incline by using v=v_0+at with the final velocity of zero plugged in. Then we compute the time for the block to slide down the ramp by using the known sliding distance and acceleration in x=1/2*at^2. We add the two times together, and we're done finding time, accelerations, sliding distance, friction coefficient for the sliding block on an incline problem!