Equation for tension in a string

Question

Two blocks are connected by a string and pulley (such that block one (m sub 1) is on an incline, block two (m sub 2) is hanging from the string), and the incline of the ramp is the angle theta. Friction is negligible. The acceleration is a = ((m2-m1 sin theta)g)/(m1+m2). This previous material is directly from a homework problem. Conceptually from there, how would I create an equation for tension?

Answer 1

In general, there are two tensions -- the tension on the string attached to m₁,

and the tension on the string attached to m₂.

Their difference is responsible for accelerating the pulley.

In the absence of any information about the pulley, I will assume that its mass is negligible.

In that case the two tensions are the same, and have some common value T.

The block m₂ experiences downward force m₂g and upward force T.

Their difference, the net force, is responsible for the acceleration a:

m₂g - T = m₂a

The block m₁ experiences downward force m₁gsin(θ) parallel to the incline, and

upward force T also parallel to the incline.

Their difference, the net force, is responsible for the common acceleration a:

T - m₁gsin(θ) = m₁a

That gives us two equations.

If you did not know the acceleration a you could calculate both a and T from those two equations,

and you would get the expression for the acceleration a that you already know.

Then you can substitute a into either of the two equations to calculate T.