Tension

claimed by Jae Hee Kim (chloejhkim)

This topic covers Tension.

What is Tension?

The tension force is the force that is transmitted through a string, rope, cable or wire when it is pulled tight by forces acting from opposite ends. The tension force is directed along the length of the wire and pulls equally on the objects on the opposite ends of the wire.

Ropes and cables are useful for exerting forces since they can efficiently transfer a force over a significant distance (e.g. the length of the rope). For instance, a sled can be pulled by a team of Siberian Huskies with ropes secured to them which lets the dogs run with a larger range of motion compared to requiring the Huskies to push on the back surface of the sled from behind using the normal force. (Yes, that would be the most pathetic dog sled team ever.) It's important to note here that tension is a pulling force since ropes simply can't push effectively. Trying to push with a rope causes the rope to go slack and lose the tension that allowed it to pull in the first place. This might sound obvious, but when it comes time to draw the forces acting on an object, people often draw the force of tension going in the wrong direction so remember that tension can only pull on an object.

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How To Calculate Tension Force

we use Newton's second law to relate the motion of the object to the forces involved. To be specific we can,

1. Draw the forces exerted on the object in question.
2. Write down Newton's second law (a= ​ΣF/m ) for a direction in which the tension is directed.
3. Solve for the tension using the Newton's second law equation (a= ​ΣF/m )

​​ We'll use this problem solving strategy in the solved examples below.

Example Problems

Example 1: Angled rope pulling on a box

A 2.0kg box of cucumber extract is being pulled across a frictionless table by a rope at an angle θ=60º as seen below. The tension in the rope causes the box to slide across the table to the right with an acceleration of 3.0 m/s^2

What is the tension in the rope?

First draw a force diagram of all the forces acting on the box.

Now we use Newton's second law. The tension is directed both vertically and horizontally, so it's a little unclear which direction to choose. However, since we know the acceleration horizontally, and since we know tension is the only force directed horizontally, we'll use Newton's second law in the horizontal direction.

1. a= ​ΣF/m (use Newtons's second law for the horizontal direction)
2. 3.0 m/s^2=Tcos60º/2.0kg ​​ (plug in the horizontal acceleration, mass, and horizontal forces)
3. Tcos60º=(3.0 m/s^2)(2.0kg)
4. T=[(3.0 m/s^2)(2.0kg)]/(cos60º)
5. T=12N

Example 2: Box hanging from two ropes

A 0.25 kg container of animal crackers hangs at rest from two strings secured to the ceiling and wall respectively. The diagonal rope under tension T1 is directed at an angle θ=30º from the horizontal direction as seen below.

What are the tensions (T1 and T2) in the two strings?

First we draw a force diagram of all the forces acting on the container of animal crackers.

Now we have to use Newton's second law. There are tensions directed both vertically and horizontally, so again it's a little unclear which direction to choose. However, since we know the force of gravity, which is a vertical force, we'll start with Newton's second law in the vertical direction.

1. a=​ΣF/m (use Newtons's second law for the vertical direction)
2. 0=(T2*sin30º-Fg)/0.25kg
3. T2=Fg/(sin30º)
4. T2=mg/(sin30º)
5. T2=[(0.25kg)(9.8m/s²)]/(sin30º)
6. T2=4.9N

Now that we know T2 we can solve for the tension T1 using Newton's second law for the horizontal direction.

1. a=​ΣF/m (use Newtons's second law for the horizontal direction)
2. 0=(T2*cos30º-T1)/0.25kg (plug in the horizontal acceleration, mass, and horizontal forces)
3. T1=T2*cos30º
4. T1=(4.9N)*cos30º
5. T1=4.2N

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References

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https://www.khanacademy.org/science/physics/forces-newtons-laws/tension-tutorial/a/what-is-tension http://www.physicsclassroom.com/Class/newtlaws/U2L2b.cfm#tension http://hyperphysics.phy-astr.gsu.edu/hbase/mlif.html http://hyperphysics.phy-astr.gsu.edu/hbase/elev.html http://www.softschools.com/formulas/physics/tension_formula/70/ http://physics.stackexchange.com/questions/36175/understanding-tension http://www.brightstorm.com/science/physics/newtons-laws-of-motion/tension/