Return to Answer

You have a few questions here. They mostly concern the energy loss due to friction. In this problem there is no energy loss because of friction.

Friction is present between the rope and the pulley. This is static friction. It makes the pulley rotate. Energy is lost due to friction (dissipated as heat) only when there is relative motion between the surfaces in contact - ie kinetic friction. In this problem we assume the rope does not slip against the pulley - there is no relative motion. So there is no work done by the rope against the force of friction, and therefore no loss of energy as the rope makes the pulley rotate.

(There might also be friction between the pulley and the axle on which it rotates. Work is done - and energy lost - in that case, because there is relative motion between the surfaces in contact - the pulley slides over the axle. But the question assumes the energy loss in this case is negligible.)

I think you are concerned that there must be energy lost because the tension is different on both sides of the pulley. These tensions are different, but this does not mean that energy is lost because the rope is doing work against friction. No work is done against friction, because the rope does not slip. The tension in the rope is doing useful work accelerating the pulley, giving it kinetic energy. This energy is not lost. (See my answer to "Tension in a string with pulley and two objects at opposite ends" http://physics.stackexchange.com/q/264798.)

The tensions in the rope and spring (and the torque on the pulley) are brought into the equations only if you are trying to solve the problem by applying F=ma. The alternative method (used by your teacher) is to apply Conservation of Energy. This method ignores the forces and looks instead at the effect they have in storing and transferring energy - ie stretching the spring or raising the mass or making the mass and pulley move faster/slower.

Back to your 1st question :

how do you find the velocity of the object after it had descended 10 cm?

The answer is : use the equation for conservation of energy which your teacher wrote. The RHS tells you how much gravitational PE the system loses when the mass descends by $h$. That is equal to the LHS, which is the elastic PE gained by the spring plus the KE gained by the pulley and the mass. Because the rope is inelastic, the extension of the spring is $x=h$. Because the rope does not slip against the pulley then $v=r\omega$. Substitute in the equation to find v, the velocity of the 1kg mass.

There is no need to solve the equation of motion to find x(t).

You have a few questions here. They mostly concern the energy loss due to friction. In this problem there is no energy loss because of friction.

Friction is present between the rope and the pulley. This is static friction. It makes the pulley rotate. Energy is lost due to friction (dissipated as heat) only when there is relative motion between the surfaces in contact - ie kinetic friction. In this problem we assume the rope does not slip against the pulley - there is no relative motion. So there is no work done by the rope against the force of friction, and therefore no loss of energy as the rope makes the pulley rotate.

(There might also be friction between the pulley and the axle on which it rotates. Work is done - and energy lost - in that case, because there is relative motion between the surfaces in contact - the pulley slides over the axle. But the question assumes the energy loss in this case is negligible.)

I think you are concerned that there must be energy lost because the tension is different on both sides of the pulley. These tensions are different, but this does not mean that energy is lost because the rope is doing work against friction. No work is done against friction, because the rope does not slip. The tension in the rope is doing useful work accelerating the pulley, giving it kinetic energy. This energy is not lost. (See my answer to "Tension in a string with pulley and two objects at opposite ends" https://physics.stackexchange.com/q/264798.)

The tensions in the rope and spring (and the torque on the pulley) are brought into the equations only if you are trying to solve the problem by applying F=ma. The alternative method (used by your teacher) is to apply Conservation of Energy. This method ignores the forces and looks instead at the effect they have in storing and transferring energy - ie stretching the spring or raising the mass or making the mass and pulley move faster/slower.

Back to your 1st question :

how do you find the velocity of the object after it had descended 10 cm?

The answer is : use the equation for conservation of energy which your teacher wrote. The RHS tells you how much gravitational PE the system loses when the mass descends by $h$. That is equal to the LHS, which is the elastic PE gained by the spring plus the KE gained by the pulley and the mass. Because the rope is inelastic, the extension of the spring is $x=h$. Because the rope does not slip against the pulley then $v=r\omega$. Substitute in the equation to find v, the velocity of the 1kg mass.

There is no need to solve the equation of motion to find x(t).

You have a few questions here. They mostly concern the energy loss due to friction. In this problem there is no energy loss because of friction.

Friction is present between the rope and the pulley. This is static friction. It makes the pulley rotate. Energy is lost due to friction (dissipated as heat) only when there is relative motion between the surfaces in contact - ie kinetic friction. In this problem we assume the rope does not slip against the pulley - there is no relative motion. So there is no work done by the system against the force of friction, and therefore no loss of energy as the rope makes the pulley rotate.

(There might also be friction between the pulley and the axle on which it rotates. Work is done - and energy lost - in that case, because there is relative motion between the surfaces in contact - the pulley slides over the axle. But the question assumes the energy loss in this case is negligible.)

I think you are concerned that there must be energy lost because the tension is different on both sides of the pulley. These tensions are different, but this does not mean that energy is lost because the rope is doing work against friction. No work is done against friction, because the rope does not slip. The tension in the rope is doing useful work accelerating the pulley, giving it kinetic energy. This energy is not lost. (See my answer to "Tension in a string with pulley and two objects at opposite ends" http://physics.stackexchange.com/q/264798.)

The tensions in the rope and spring (and the torque on the pulley) are brought into the equations only if you are trying to solve the problem by applying F=ma. The alternative method (used by your teacher) is to apply Conservation of Energy. This method ignores the forces and looks instead at the effect they have in storing and transferring energy - ie stretching the spring or raising the mass or making the mass and pulley move faster/slower.

Back to your 1st question :

how do you find the velocity of the object after it had descended 10 cm?

The answer is : use the equation for conservation of energy which your teacher wrote. The RHS tells you how much gravitational PE the system loses when the mass descends by $h$. That is equal to the LHS, which is the elastic PE gained by the spring plus the KE gained by the pulley and the mass. Because the rope is inelastic, the extension of the spring is $x=h$. Because the rope does not slip against the pulley then $v=r\omega$. Substitute in the equation to find v, the velocity of the 1kg mass.

There is no need to solve the equation of motion to find x(t).

You have a few questions here. They mostly concern the energy loss due to friction. In this problem there is no energy loss because of friction.

Friction is present between the rope and the pulley. This is static friction. It makes the pulley rotate. Energy is lost due to friction (dissipated as heat) only when there is relative motion between the surfaces in contact - ie kinetic friction. In this problem we assume the rope does not slip against the pulley - there is no relative motion. So there is no work done by the rope against the force of friction, and therefore no loss of energy as the rope makes the pulley rotate.

(There might also be friction between the pulley and the axle on which it rotates. Work is done - and energy lost - in that case, because there is relative motion between the surfaces in contact - the pulley slides over the axle. But the question assumes the energy loss in this case is negligible.)

I think you are concerned that there must be energy lost because the tension is different on both sides of the pulley. These tensions are different, but this does not mean that energy is lost because the rope is doing work against friction. No work is done against friction, because the rope does not slip. The tension in the rope is doing useful work accelerating the pulley, giving it kinetic energy. This energy is not lost. (See my answer to "Tension in a string with pulley and two objects at opposite ends" http://physics.stackexchange.com/q/264798.)

The tensions in the rope and spring (and the torque on the pulley) are brought into the equations only if you are trying to solve the problem by applying F=ma. The alternative method (used by your teacher) is to apply Conservation of Energy. This method ignores the forces and looks instead at the effect they have in storing and transferring energy - ie stretching the spring or raising the mass or making the mass and pulley move faster/slower.

Back to your 1st question :

how do you find the velocity of the object after it had descended 10 cm?

The answer is : use the equation for conservation of energy which your teacher wrote. The RHS tells you how much gravitational PE the system loses when the mass descends by $h$. That is equal to the LHS, which is the elastic PE gained by the spring plus the KE gained by the pulley and the mass. Because the rope is inelastic, the extension of the spring is $x=h$. Because the rope does not slip against the pulley then $v=r\omega$. Substitute in the equation to find v, the velocity of the 1kg mass.

There is no need to solve the equation of motion to find x(t).

You have a few questions here. They mostly concern the energy loss due to friction. In this problem there is no energy loss because of friction.

Friction is present between the rope and the pulley. This is static friction. It makes the pulley rotate. Energy is lost due to friction (dissipated as heat) only when there is relative motion between the surfaces in contact - ie kinetic friction. In this problem we assume the rope does not slip against the pulley - there is no relative motion. So there is no work done by the system against the force of friction, and therefore no loss of energy as the rope makes the pulley rotate.

There might also be friction between the pulley and the axle on which it rotates. Work is done (and energy lost) in that case, because there is relative motion between the surfaces in contact - the pulley slides over the axle. But the question assumes the energy loss in this case is negligible.

The tensions in the rope and spring (and the torque on the pulley) are taken into account only if you are trying to solve the problem by applying F=ma. The alternative method (used by your teacher) is to apply Conservation of Energy. This method ignores the forces and looks instead at the effect they have in storing energy eg stretching the spring or raising the mass or making the mass and pulley move faster/slower.

Back to your 1st question :

how do you find the velocity of the object after it had descended 10 cm?

The answer is : use the equation for conservation of energy which your teacher wrote. The RHS tells you how much gravitational PE the system loses when the mass descends by $h$. That is equal to the LHS, which is the elastic PE gained by the spring plus the KE gained by the pulley and the mass. Because the rope is inelastic, the extension of the spring is $x=h$. Because the rope does not slip against the pulley then $v=r\omega$. Substitute in the equation to find v, the velocity of the 1kg mass.

There is no need to solve the equation of motion to find x(t).

You have a few questions here. They mostly concern the energy loss due to friction. In this problem there is no energy loss because of friction.

Friction is present between the rope and the pulley. This is static friction. It makes the pulley rotate. Energy is lost due to friction (dissipated as heat) only when there is relative motion between the surfaces in contact - ie kinetic friction. In this problem we assume the rope does not slip against the pulley - there is no relative motion. So there is no work done by the system against the force of friction, and therefore no loss of energy as the rope makes the pulley rotate.

(There might also be friction between the pulley and the axle on which it rotates. Work is done - and energy lost - in that case, because there is relative motion between the surfaces in contact - the pulley slides over the axle. But the question assumes the energy loss in this case is negligible.)

I think you are concerned that there must be energy lost because the tension is different on both sides of the pulley. These tensions are different, but this does not mean that energy is lost because the rope is doing work against friction. No work is done against friction, because the rope does not slip. The tension in the rope is doing useful work accelerating the pulley, giving it kinetic energy. This energy is not lost. (See my answer to "Tension in a string with pulley and two objects at opposite ends" http://physics.stackexchange.com/q/264798.)

The tensions in the rope and spring (and the torque on the pulley) are brought into the equations only if you are trying to solve the problem by applying F=ma. The alternative method (used by your teacher) is to apply Conservation of Energy. This method ignores the forces and looks instead at the effect they have in storing and transferring energy - ie stretching the spring or raising the mass or making the mass and pulley move faster/slower.

Back to your 1st question :

how do you find the velocity of the object after it had descended 10 cm?

The answer is : use the equation for conservation of energy which your teacher wrote. The RHS tells you how much gravitational PE the system loses when the mass descends by $h$. That is equal to the LHS, which is the elastic PE gained by the spring plus the KE gained by the pulley and the mass. Because the rope is inelastic, the extension of the spring is $x=h$. Because the rope does not slip against the pulley then $v=r\omega$. Substitute in the equation to find v, the velocity of the 1kg mass.

There is no need to solve the equation of motion to find x(t).