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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.

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.

Yes. Tension can vary if external forces are acting on 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 attach a mass m at B somewhere between A and C. Section BC is still supporting the same mass M, so the tension in section BC remains Mg. 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.

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.

Yes. Eqn (iii) can be written as :
$T_1 - T_2 = \frac{I\alpha}{R^2}$.

$T_1 \ne T_2$ if :
(i) the pulley is not massless ($I\ne0$);
(ii) the string does not slip against the rim of the pulley (so $a=R\alpha$) and the hanging masses are not equal ($M_1 \ne M_2$) so that $a \ne 0$ hence $\alpha \ne 0$.

Yes. Tension can vary if external forces are acting on 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 attach a mass m at B somewhere between A and C. Section BC is still supporting the same mass M, so the tension in section BC remains Mg. 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.