How to Calculate Tension

Tension is the force that a line or string exerts on a load. By Newton’s third law of motion, it is also the force the load exerts on the line. The main strategy to such problems is often to look at the bodies in isolation of the others and assess all the forces acting on it.

Instructions

1. Draw a setup where a pulley extends from the edge of a table. A line drapes over the pulley, with a hanging weight of 5 Newtons on one end and a 3-Newton weight sitting on the table at the other end of the line. A Newton is the SI unit for force. Suppose that the pulley has an elevation such that the line to the body on the table is vertical. Suppose further that the body on the table rests on a track of rollers (for example, straws), so that the coefficient of friction between the two is negligible, so that you can concentrate just on the tension.

2. Examine the forces on the hanging weight separately. Gravity pulls it down with a force of 5 Newtons (N), and equals the body’s mass times the gravitational acceleration constant, 9.80 meters per second-squared (m/s^2). But the block on the table slows the gravitational acceleration, via the tension in the line. Denote this tension with the letter T.
   Then the total force accelerating the body equal F=ma=mg-T, where you subtract T because it’s less than mg and you want acceleration a to be positive, for simplicity. Mass m is just the body’s weight divided by g, in other words 5N/9.8m/s^2 = 0.510kg. So the equation becomes 0.510xa=5N-T.

3. The body on the table is only being pulled by tension T. So ma=T. The mass m is 3/9.8 =0.306kg. So 0.306a=T. You now have two equations in two unknowns.

4. Solve for acceleration a in both equations, since the two bodies and string accelerate together, at the same rate. So T/0.306 = a = (5N-T)/0.510. So you can eliminate acceleration a, giving you T/0.306 = (5N-T)/0.510.

5. Solve for tension T, using basic arithmetic. In the example above, the result is T=1.88N.

References

• "Fundamentals of Physics"; David Halliday and Robert Resnick; 1992
• Johns Hopkins U.: Tension in a String—Atwood’s Machine
• Georgia State U.: Atwood’s Machine