Tension in a string with pulley and two objects at opposite ends

Question

Tension is confusing me. According to my understanding of tension the tension in a string is uniform but while going through the solution of a problem I read that tge tension on the left part of the string is T1 and thst on the right side of the string is T2.
Can a single string have two different tensions?
Please refer to the image for further explanation of doubt.

Answer 1

For a pulley that has mass and moment of inertia, there must be a net torque on the pulley for the pulley to demonstrate an angular acceleration. Assuming that mass "M" is greater than mass "m", a net torque necessarily requires that the counterclockwise torque from mass "M", given by the equation Torque1 = T1(R), is larger than the clockwise torque from mass "m", given by the equation Torque2 = -T2(R). Since the term "R" is common to both torques, it is readily apparent that tension T1 must be greater than tension T2.

Answer 2

Yes. Tension can vary if external forces are acting between the ends of the string - such as gravity (if the string has mass) and friction where the string makes contact with other objects (such as the pulley).

For example, suppose you attach one end A of a uniform massless string to a support and the other end C to a vertically hanging mass M. This creates uniform tension Mg in the string everywhere between A and C.

Now tie a mass m to point B on the string somewhere between A and C. Section BC is still supporting the same mass M, so the tension in section BC remains Mg. However, section AB is now supporting a mass M+m, so the tension in section AB increases to (M+m)g. AB and BC are still part of the same string, but the tension in each section differs.

Tension can also vary along a string which has non-zero mass and is being accelerated. Acceleration is equivalent to gravity.

In the illustration, the string is in contact with a pulley. Friction between the string and pulley opposes relative motion. This is similar to adding the weight at B between the ends of the string AC.

Answer 3

The tension in a string is uniform

Not always.

The tension of the string is uniform in some cases:

1. If String is mass-less and its particles don't move with respect to each other (i.e. string is inextensible or if it is extensible, it reach to its final tension).

2. If String is mass-less and there is no friction between string and pulley.

3. For string with mass, if it is fixed, then tension will be uniform.

In your case, option 2 has been violated (there is friction between pulley and string).