# Tension of a rope with hanging mass and acceleration relationship

### What is tension? (article) | Tension | Khan Academy

The reason tension in the rope exists is due to the inertia of m1 resisting the Take your first equation and rearrange for force: In your example the only quantity that should be equivalent between both masses is the acceleration. Total tension in a rope caused by two hanging masses at opposite ends?. In physics, tension is the force exerted by a rope, string, cable, or similar In this case, we must account for the acceleration on the weight as well as the via the following equation: Force due to friction (usually written Fr) = (mu)N, For a system of two masses hanging from a vertical pulley, tension equals. (Forces perpendicular to the slope add to zero, since there is no acceleration in that find a rubber band, some objects to hang from the end of the rubber band, Tension in the rope must equal the weight of the supported mass, as we can . Observing [link], we can use trigonometry to determine the relationship between.

This was the technique used by Galileo in his pioneering studies of motion under gravity--by diluting the acceleration due to gravity, using inclined planes, he was able to obtain motion sufficiently slow for him to make accurate measurements using the crude time-keeping devices available in the 17th Century. Block sliding down an incline Consider two masses, andconnected by a light inextensible string.

Suppose that the first mass slides over a smooth, frictionless, horizontal table, whilst the second is suspended over the edge of the table by means of a light frictionless pulley. Since the pulley is light, we can neglect its rotational inertia in our analysis. Moreover, no force is required to turn a frictionless pulley, so we can assume that the tension of the string is the same on either side of the pulley.

## Tension Formula

Let us apply Newton's second law of motion to each mass in turn. The first mass is subject to a downward forcedue to gravity.

However, this force is completely canceled out by the upward reaction force due to the table. The mass is also subject to a horizontal forcedue to the tension in the string, which causes it to move rightwards with acceleration The second mass is subject to a downward forcedue to gravity, plus an upward force due to the tension in the string.

These forces cause the mass to move downwards with acceleration Now, the rightward acceleration of the first mass must match the downward acceleration of the second, since the string which connects them is inextensible.

Thus, equating the previous two expressions, we obtain Note that the acceleration of the two coupled masses is less than the full acceleration due to gravity,since the first mass contributes to the inertia of the system, but does not contribute to the downward gravitational force which sets the system in motion. Block sliding over a smooth table, pulled by a second block Consider two masses, andconnected by a light inextensible string which is suspended from a light frictionless pulley, as shown in Fig.

**Physics Mechanics - Pulley With Two Hanging Masses, Calculate Acceleration & Tension Force**

Let us again apply Newton's second law to each mass in turn. Without being given the values of andwe cannot determine beforehand which mass is going to move upwards. Let us assume that mass is going to move upwards: The first mass is subject to an upward forcedue to the tension in the string, and a downward forcedue to gravity.

These forces cause the mass to move upwards with acceleration The second mass is subject to a downward forcedue to gravity, and an upward forcedue to the tension in the string. These forces cause the mass to move downward with acceleration Now, the upward acceleration of the first mass must match the downward acceleration of the second, since they are connected by an inextensible string.

Hence, equating the previous two expressions, we obtain As expected, the first mass accelerates upward i. Note that the acceleration of the system is less than the full acceleration due to gravity,since both masses contribute to the inertia of the system, but their weights partially cancel one another out. For objects where the mass is distributed at different distances from the axis of rotation, there is some multiplying factor in front of the MR2.

Newton's second law for rotation You can figure out the rotational equivalent of any straight-line motion equation by substituting the corresponding rotational variables for the straight-line motion variables angular displacement for displacement, angular velocity for velocity, angular acceleration for acceleration, torque for force, and moment of inertia for mass.

Try this for Newton's second law: Replace force by torque, m by I, and acceleration by angular acceleration and you get: Example - two masses and a pulley We've dealt with this kind of problem before, but we've never accounted for the pulley.

## What is tension?

There are two masses, one sitting on a table, attached to the second mass which is hanging down over a pulley. When you let the system go, the hanging mass is pulled down by gravity, accelerating the mass sitting on the table. When you looked at this situation previously, you treated the pulley as being massless and frictionless. We'll still treat it as frictionless, but now let's work with a real pulley.

A N block sits on a table; the coefficient of kinetic friction between the block and the table is 0. This block is attached to a N block by a rope that passes over a pulley; the second block hangs down below the pulley.

The pulley is a solid disk with a mass of 1. The rope passes over the pulley on the outer edge. What is the acceleration of the blocks? As usual, the first place to start is with a free-body diagram of each block and the pulley. Note that because the pulley has an angular acceleration, the tensions in the two parts of the rope have to be different, so there are different tension forces acting on the two blocks.

For block 1, the force equations look like this: For block 2, the force equation is: The pulley is rotating, not moving in a straight line, so do the sum of the torques: Just a quick note about positive directions Block 1 accelerates right, so make right the positive direction for block 1, and for the pulley, which will have a clockwise angular acceleration, make clockwise the positive direction.

We have three equations, with a bunch of unknowns, the two tensions, the moment of inertia, the acceleration, and the angular acceleration. We expect this mass to have an acceleration a that is up.

### Torque and rotational inertia

There are no horizontal forces. We will take up as positive. So we need more inormation. We expect this mass to have an acceleration a that is down.

We can choose to call down "positive" for this mass or we can call up "positive" and then we expect this mass to have an acceleration of - a. Either choice is fine. This time, let's choose down as "positive".

### A pulley system — Collection of Solved Problems

But now we have two equations with two unknowns and that is sufficient. Now, let's consider an inclined Atwoods machine.

Masses m1 and m2 are connected by a string which runs over a pulley and mass m2 sits on a smooth inclined plane.