An elevator is a box that moves up and down a building with the help of strong cables. The tension in the cables enables the elevator to move heavy weights upward and downward. The counterweights provide tension for the cables to hold the elevator. The entire movement is controlled by intelligent electronic systems. While entering an elevator you might have noticed a notice mentioning the maximum number of persons allowed. This value is calculated by using the formula of tension and the strength of the cables. In this article, we will calculate the formula for tension in the cables of an elevator at three conditions: rest, moving upwards and moving downwards.
What is weightlessness?
We feel our weight due to the reaction force given by the support (ground) that we stand on. But, when there is no reaction force acting on use we don’t feel our weight. This is experienced in high-speed elevators or in worst cases when the cables of the elevator break. Other examples of weightlessness include astronauts on the moon, paragliding, etc.
Tension in the cables of an elevator
Tension is a pulling force that acts in one dimension along the axis of the cables opposite to the direction of the force applied. In the case of an elevator, the pulling force in the cables is provided by the combined weight of the elevator box and the person traveling inside it. So, ideally, the tension in the cable should be equal to the weight of the elevator at a state of rest. Most modern elevators use composite steel cable wires for better strength.
The formula for tension in a cable is not equal to the weight of the elevator all the time. It depends on whether the elevator is moving or not, and if it is moving then in which direction? Let us derive the formula for tension in three different cases.
The formula for tension in cables of the elevator when at rest
When an elevator is at rest the weight of the elevator plus the person inside it is borne by the tension in the cables. The formula for tension will be straightforward: T= m*g
The formula for tension in cables of the elevator when moving upwards
When an elevator is moving upwards, it is moving against the force of gravity, hence a greater force is required to pull the elevator up. Let us derive the formula using the free-body diagram above. The forces acting on the elevator are Tension and weight of the elevator plus the weight of the person. The net force (m*a) should be equal to the sum of forces acting on the elevator.
T – mg = ma
T = mg + ma = m(g+a)
Hence, from the formula, we can see the tension value will be greater compared to when at rest.
The formula for tension in cables of the elevator when moving downwards
When an elevator is moving downwards, it is moving in the direction of the force of gravity, hence a lesser force is required compared to the state of rest. Let us derive the formula using the free-body diagram above. The forces acting on the elevator are Tension and weight of the elevator plus the weight of the person. The net force (m*a) should be equal to the sum of forces acting on the elevator.
mg – T = ma
T = mg – ma= m(g-a)
Hence, from the formula, we can see that the tension value will be less compared to when at rest.
Problems with tension in the cables of an elevator
An elevator with a weight of 110 Kg is carrying 2 people with weights of 40Kg and 50Kg. Calculate the tension in the cable of the elevator at a)rest b)moving upwards with an acceleration of 5 m/s2 c)moving downwards with an acceleration of 5 m/s2
a) When elevator is at rest: T = mg
Total weight = 110 + 40 +50 =200 Kg
Tension = 200*9.8 = 1960 N
b) When elevator is moving upwards: T = mg + ma
Total weight = 110 + 40 +50 =200 Kg
Tension = 200*9.8 + 200*5 = 2960 N
b) When elevator is moving downwards: T = mg – ma
Total weight = 110 + 40 +50 =200 Kg
Tension = 200*9.8 – 200*5 = 1196 N
What happens when acceleration(a) is equal to the acceleration due to gravity(g)?
When acceleration(a) is equal to the acceleration due to gravity(g) then it becomes a case of free-fall. This might be due to the breaking of the cables. In this case, according to the above formula T= mg- ma, the value of tension becomes zero.
What happens when acceleration(a) is greater than the acceleration due to gravity(g)?
Such a case is not possible under normal conditions. In this case, according to the above formula T= mg- ma, the value of tension becomes negative. That means an external force is acting on the elevator in a downward direction.
See also
- Tension formula- Tug of war
- Tension Formula-Tension in a rope pulling blocks horizontally
- Tension formula-Rope pulling blocks horizontally with kinetic friction involved
- Tension formula-Rope
- Tension formula: Tension in a vertically suspended wire with a weight
- Formula For Tension
- Tension formula circular motion
- Pulley system
