Two blocks are connected by string. The inclination of ramp is 39 degrees while the mass of blocks are m1 = 8.8 kg and m2 = 16.4 kg. Coefficient of kinetic friction of inclined block and ramp is 0.38

Question

a) Calculate the magnitude of the acceleration a of the system in m/s^2.

b) Calculate the magnitude of tension T in the string in Newtons.

Answer 1

The situation is as follows : two blocks attached by a rope , of masses m (in front) and M (behind) , slide down an incline of angle θ. There is friction between the surface of the incline and the block; the coefficient of kinetic friction is μ. We wish to determine the rate of acceleration of this composite object down the incline.

Let us focus on any one of the blocks, and list out the forces acting on it.

The first force is just the force due to gravity. This is just mg, acting vertically downward.

Note however, that the reason we are interested in finding forces acting on a body is that force causes motion, and we are ultimately interested in knowing how bodies are moving (or not moving). In this problem, the block cannot move vertically down, or in its ⊥ direction, the horizontal; instead, it is constrained to move on the surface of the incline, tilted at θ w.r.t the horizontal. So we need to resolve forces along the incline, and ⊥ to it. The force due to gravity has two components : mg sin θ down the incline, and mg cos θ acting normally (i.e. ⊥) into the surface of the incline.

Newton's 3rd law: If the block pushes down on the surface of the incline with a force mg cos θ, then the surface of the incline pushes back against the block with a 'Normal' force N = mg cos θ.

Next, there is a force due to friction. Kinetic friction is the form of friction that is active when the block is in motion. It depends on the mutual pushing between the two surfaces, which is just the normal force N. Also, friction always opposes the sense in which motion is happening. So, the force due to friction is μ N acting up the incline.

Finally, we have tension in the rope attaching the two blocks. This is some T acting towards the other block. So that m experiences T going up the incline (towards M), whereas M experiences T acting down the incline.

The blocks slide down the incline with an acceleration a. For the first block, we may write :

mg sin θ - μ N_m - T = ma

where N_m is the normal force experienced by m, and for the second block, we may write :

T + Mg sin θ - μ N_M = Ma

where N_M is the normal force experienced by M. There are two equations and two unknowns; so they can be solved to find the acceleration a and the tension T in terms of the knowns (m , M , .θ , μ)

Hope this helps.