Question

1. A 4 kg mass is resting on a ramp inclined at an angle of 30 degrees with respect to the horizontal. A string is attached to the 4 kg mass and passes over a frictionless pulley and is connected to a 5 kg mass hanging over the side. If the coefficient of friction between the ramp and the 4 kg mass if 0.3 find; A) The acceleration of both masses B) The tension in the string C) Find the speed of both blocks after they move a distance of 2 meters.

Answer 1

Let's see an image of the situation, first we set the reference system with the positive directions. In the image we see the forces acting of each body then we set the equations in the x and y axis for both bodies. The the sense of the acceleration have been assumed if the result turns out to be negative the sense is opposite to the chosen but the magnitude is the same:

For the 5 kg body we have:

(1)

In the x-axis there is no motion and there are no forces. In the y-axis acts the tension and the weight, the acceleration points to the negative sense in the reference system.

For the 4 kg body:

(2)

(3)

The friction force is proportional to the normal force as follows:

Substituting in the y-axis equation.

Since the string is the same for both bodies we solve for T , (1) and (3) are equal:

(4)

We substitute (2) in (4):

The terms that depend on the acceleration are grouped in one side of the equation:

We solve for a:

Substituting values it yields:

Substituting the acceleration value in (1):

The velocity in an accelerated body can be written as:

Since the blocks are at rest at first: