Ropes can be used to pull heavy objects. Whenever we pull a rope, tension is developed in the rope. The formula for tension is simple: when we consider one weight attached to the rope, it is equal to the force applied. The formula for tension becomes complex when we attach more than one weight. In all the below cases, we assume frictional force to be negligible.
Formula for tension
First, let us calculate the net acceleration for the system by using the formula for force:
Fnet = Total mass*acceleration F = (M1 + M2)*a Acceleration(a) = F/(M1+M2)
Tension equation for block 1
Now we will solve the equation for Block1. Using the free-body diagram we find that the forces acting on block1 is just tension which is in the same direction as the net force. Net force will be equal to the mass times the acceleration. The acceleration will be the same as the net acceleration of the system as each block will move with the same acceleration. But the net force on the block will be different as compared to the system.
Fnet = T
M1*a = T
T = M1*a —————————(1)
Tension equation for block 2
Now let us solve the equation for Block2. Using the free-body diagram again, we find that the tension force and the applied force F are acting in opposite directions. The summation of both forces should be equal to the net force. Again the acceleration of the block will be the same as that of the system, but the net force on the block will be different as compared to the system.
Fnet = F – T
M2*a = F – T
T = F + M2*a —————————-(2)
The two values of tension from eq(1) and eq(2) should be equal. This can be used to verify the answer.
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Numerical problem on Tension in a rope pulling blocks horizontally
A force of 50 N is applied to pull the two-block attached by a rope. Calculate the tension in the rope.
First, let us calculate the acceleration of the system by using the formula for force. The unit for force is in newtons so the unit of acceleration here will be m/s2.
Fnet = Total mass*acceleration Acceleration = 50/5 = 10 m/s2
Now, the net force action on block with weight 2 Kg will be: mass*acceleration = 2*10 = 20 N
Now equating the forces acting on the block to the net force. (Only tension force is acting on the block with weight 2 Kg and is in the same direction of that of the net force, so the sign will be positive)
Fnet = T
20 = T
T = 20 N
The value of tension is 20 N, it will be the same throughout the rope according to our assumption of a massless rope. Now, for verification let us calculate the tension force using the formula for the other block.
The net force action on block with weight 3 Kg will be: mass*acceleration = 3*10 = 30 N
Now equating the forces acting on the block to the net force. (The tension force and the force of 50 N are acting on the block in opposite directions.)
Fnet = 50 – T
30 = 50 – T
T = 20 N
Hence we have proved that from both the formulas for the tension we arrive at the same value. This proves the validity of our formula for tension in a rope pulling blocks horizontally.
