Tension is the force that is exerted through the length of a rope or string or wire or cables. There is no specific formula for tension. Tension is a type of contact force. The basic laws of physics can be applied to calculate the tension force in a string or rope. Tension is always a pulling force and is exerted in a rope only when it is pulled by a force. Here, in this article, we discuss the formula for tension in different scenarios.
Formula For Tension:
While solving tension-related numerical problems, we consider certain assumptions without which it is difficult to calculate tension using the basic laws of physics. This can be considered an ideal case which is not always true in real-life scenarios.
- The tension is the same in all points of the rope.
- The rope under consideration is massless.
- Ignore the minor effects of friction, air resistance, and other undesirable factors unless stated specifically.
- All calculations are done on planet Earth and the value of g (acceleration due to gravity is 9.8 m/s2.
In this article, we calculate the formula for tension for the following 10 scenarios:
- Tension in a rope pulling blocks horizontally
- Tension in a rope pulling blocks horizontally with kinetic friction involved
- Tension in a rope during Tug of war
- Tension in vertically suspended wire with a weight
- Tension in a rope attached to a weight at an angle
- Tension in an elevator
- Man walking on a tight rope
- Tension in a wire under circular motion
- Tension in the rope of a pulley
- Tension in a wire with an inclination and pulley
The formula for tension in a rope pulling blocks horizontally
This is the most common form of tension in a string problem. A rope is used to pull two blocks separated by a distance. There can be n number of blocks, but the tension in each rope will be different. Here, a force F is pulling blocks with mass M1 and M2 across a frictionless surface(uk=0).
For the entire system: Fnet (for the whole system) = Total mass*acceleration => Acceleration(a) = F/(M1+M2)
Using the free body diagram as shown above we focused on the two blocks separately and calculated the formula for tension.
T = M1*a T = F + M2*a
Both equations can be used to find the value of tension in the rope. The values of tension will be the same. The SI unit of tension force is Newton. You can read our blog post on units of force for more details.
The formula for tension in a rope pulling blocks horizontally with kinetic friction involved
Kinetic friction is the opposing force between two bodies in relative motion. In an ideal case of tension problems, there will always be a friction factor involved. We generally skip it for the sake of making the calculations simpler. There will be static friction involved for cases where the force applied to the system is less than limiting friction.
Here the net force of the system will involve the pulling force applied, tension force plus the frictional force in the opposite direction to the force applied. The point to be noted is frictional force will apply to each and every object with mass.
The formula for friction = μk*N => μk * (m*g) [where μk is the coefficient of kinetic friction] Friction of the entire system = μk(m1+m2)*g Friction of block1 = m1*g Friction of block2 = m2*g
Using free body diagram as shown above,
Block1: There is a tension force and frictional force in opposite directions.
Fnet = T – friction => T = μk*m1*g + m1*a
Block 2: There is an applied force towards the right. The tension and frictional force are acting in the same direction.
Fnet = F – T – friction => T = F – μk*m2*g – m2*a
The formula for tension in a rope during Tug of war
A game of Tug of war involves two teams pulling a single rope towards each other in opposite directions. The team that pulls the opponent towards its side wins. The force of the team is transferred through the tension of the rope to pull the opponent. Let us find the formula for tension in such a scenario.
F = mass x acceleration => Fnet = total mass x anet => anet = Fnet /total mass
The acceleration in the whole system will be the same. We use this value of acceleration in the equation of forces for both bodies and calculate the formula for tension. You can read here for a detailed derivation.
T = F1 – [(M1+M2)*acceleration] T = F2 – [(M3+M4)*acceleration]
The formula for tension in vertically suspended wire with a weight
The formula for tension in a rope attached to a weight at an angle
Strings will not be always attached in a vertical direction or horizontal position. They are generally attached at an angle according to convenience. For example, the serial lighting wires are made to hang in a Christmas tree freely. You may have seen the cloth hanging line which is never straight, it bends down due to the weight. We will try to calculate the tension in the strings of such a similar case.
Here, a weight (mass m) is suspended with the help of two wires with tension T1 and T2. Since there are two wires involved the tension can be different. The angle the wires make with the ceiling is given as a and b.
First, we will draw the free body diagram for the block. The tension forces are resolved in the horizontal and vertical directions. Force mg will act in the downward direction due to gravitational pull, and tension will act in the opposite direction i.e. upward.
T1 sin(a) + T2 sin(b) = m*g ———-(1)
T1cos(a) = T2cos(b)———————(2)
Solving equations (1) and (2) we can arrive at a solution for T1 and T2. Note that if the angle a=b then the tension T1 will be equal to T2.
The formula for tension in an elevator
This is a classic numerical problem in physics for tension force. When anybody is suspended by a rope in a state of rest, then the tension is equal to the weight of the body. But when the rope is pulled up or down by an acceleration then the value of tension changes. The tension will be always higher when pulling the body up as compared to going down.
T = m*g [ When elevator is at rest]
T = m*(g-a) [ When elevator is going down ]
T = m*(g+a) [When elevator is moving upwards]
Man Walking on a tightrope
You might have seen or heard of stuntmen walking on a tightrope. This problem is similar to a weight suspended by two wires. The weight of the man can be assumed as a block that is suspended freely by two wires forming an angle with the ceiling. The tension will be different in the two sections of the rope except for when the person is exactly at the mid-position.
T1 sin(a) + T2 sin(b) = m*g ———-(1)
T1cos(a) = T2cos(b)———————(2)
The formula for tension in a wire under circular motion
Have you ever swung any weight attached to a string? You would have felt the tension of the string in your hands.
Here, the force mg of the weight due to gravity will act downward and the tension will act towards us, away from the weight. There is one more additional force acting on the weight. This is called the centripetal force. This is the force that tries to pull the object towards the center i.e. toward you. So, there are three forces acting on the body gravity, tension, and centripetal force.
We have three vectors, so we solve the vectors and find the formula for tension in the string.
Tension = sqrt[(mg)2+(mv2/r)2]
The formula for tension in the rope of a pulley
Pulley is used to reducing the effort in lifting heavy objects. It consists of a fixed wheel with a rope or a string running over it freely. While solving for tension force we assume the friction between the rope and pulley wheel to be zero.
There are two cases in a pulley system. The rope can be moving up or down with an acceleration a. The tension in both segments will be equal. We write the equation for net force in each of the blocks and calculate the tension in the rope from the two equations.
For block1: moving upward against gravity at an acceleration of a
m1*a = T – m*g [net force = tension – force due to gravity]
T = m1*a + m1*g
For block2: moving downward with gravity at an acceleration of a
m1*a = m*g – T [net force = force due to gravity – tension]
T = m1*g – m1*a
The formula for tension in a wire with an inclination and pulley
In real life, the force exerted on an object will be perfectly horizontal or vertical. It will always be at an angle. So, let us look at this example where two blocks are attached to the rope of a pulley. One block is placed at an incline which will also involve kinetic friction with the surface.
The block m1 will have four forces acting on it. The weight, normal force, tension, and kinetic frictional force. We have to resolve the forces to form an equation in the horizontal and vertical directions.
Net force(M₁) = T – m₁gsinθ – friction T = M₁a – M₁gsinθ – μkM₁g
The block m2 will have two forces acting on it. The weight and the tension of the rope.
Net force(M₂) = M₂g – tension T = M₂g – M₂a
Using the above two equations we can solve for the value of tension force.
Research on Tension Force:
The tension of the strings in a guitar can affect the sound that it produces. The tension of the string is adjusted while tuning the guitar to get a particular note. Read this article on “The Effect of String Tension Variation on the Perceived Pitch of a Classical Guitar” to learn more about this.
Can tension have a negative value?
If the value of tension is negative then it basically means the force is in the opposite direction. Secondly, since the force is in opposite direction, in most cases, there might be a compressive force acting on the rope instead of a tension force.
Is the tension value the same at all points of the rope?
For the sake of simplicity, we assume the rope to be massless during calculation. In reality, the rope will have some mass. And, the tension value in the rope will be different at different locations on the rope.
Suppose for example you are pulling a rope from a ceiling with force F1, then tension at the end of the rope nearer to you will be roughly F1. But, as you move further, the value of tension will decrease.
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
- Tension elevator
- Tension formula circular motion
- Pulley system
