In this article, we will discuss what is negative tension, when it comes into the picture, and how to find it along with examples.
The negative tension comes in the act if the tensional force across the string is less than the weight of the mass attached to it. This is also true, that the tension is acting across the string, making action-reaction pair, so if we consider positive tension in the positive axis then the tension in the opposite direction has to be a negative tension.
What is Negative Tension?
The tension is exerted all across the spring, rope, or strings, and varies depending upon the mass, position, and types of forces experienced on the object attached to it.
If the influence of the tension of the string on the object is less as compared to the weight of the object the string is attached to, that is W>T, then the tension on the string is negative.
Let us understand a valid condition for tension to be negative and how it is different from the other examples. Positive tension is act exactly opposite to the weight of the object attached to it if the object is fixed at a point.
Consider an object of mass ‘m’ attached to a string. A force due to the weight of the mass is acting downward and hence the tension on the string is acting upward across the string.
The net force on the object is
F=T-mg
A force due to weight is acting in a negative y-direction, hence the negative sign.
Now, the acceleration of the object is also in the negative y-direction as the object is accelerating downward.
Hence, we have,
-ma=T-mg
Therefore, the tension in the string is
T=mg-ma
T=m(g-a)
From the above equation, we can say that the tension is negative if a>g. But, this is not a valid case. Let us see further in this article, in which situations we can find negative tension.
Read more on Tension Between Two Blocks: Several Entities And Problem Examples.
When Tension is Negative?
The tension is imposed on the string in the direction opposite to the force acting due to the weight of the object.
The negative tension can be considered as the acceleration of the object due to compression force, a condition where the weight of the object and the tension, both are exerted in the same direction.
The tension will be negative in the following three cases that we are going to discuss below.
Case 1: When a body is accelerating down
Consider an object of mass ‘m’ attached to a string accelerating downward. The tension on the string is also acting in the negative y-axis direction.
The equation of force for the above diagram is
F=-T-mg
-ma=-T-mg
T=m(a-g)
If a<g or a=0,
Then T=-ve or T=-mg
Case 2: When a body is accelerating upward
Consider an object of mass ‘m’ accelerating upward. The tension on the string is also acting downward.
The equation of force for the above diagram is
F=-T-mg
ma=-T-mg
T=-m(a+g)
Here, in this case, the tension is clearly negative.
Case 3: An object in a vertical axis with zero acceleration
Consider an object accelerating in the vertical axis with the help of a rope. The object will experience centripetal force. Let us draw a free-body diagram for it.
The force experienced on the object in a centripetal motion is F=mv2/r which is equal to the tension on the rope if force due to weight is absent.
At a certain point while turning the object is felt heavier, during that time the tension in a rope is equal to the sum of centripetal force acting on the object and the weight of the object. At some point the object feels lighter that is when the tension acts outward, hence the equation of force becomes
F=T+mgSinθ
mv2/r=T+mgSinθ
Hence, the equation for tension becomes,
T=mv2/r-mgSinθ
A tension is negative if v=0 and θ=900, that is the accelerating object stops at 90 degree angle.
Read more on How To Calculate Tension In A String:Exhaustive Insights.
Can Tension be Negative?
A tension is positive when the force is applied to pull the object with the help of a string, or rope.
If instead of pulling, a compressive force is applied, then the tension on the string can be negative. This could also be a case when the strength of the tension on the string is less compared to the weight attached to it.
This is also categorized as the compression force on the string. But string or rope can’t be compressed; only the spring can be compressed. Hence, on the application of the compressive force, the tension on the spring is negative.
Read more on Is tension a conservative force: Exhaustive Insight.
How to Find Negative Tension?
Negative tension is simply a compression force and acts always in the direction of the weight of the object.
The negative tension in the spring can be calculated by measuring the net force imposed on the object and then finding the acceleration of the object due to force.
Let us understand how to calculate the negative tension by solving the problem given below.
Problem: An object of mass 200 grams attached to a spring is compressed due to which the acceleration of the object is found to be 1m/s. Find the tension in the string.
Given: m=200 grams = 0.2kg
a=1m/s
g=9.8m/s2
Let us first draw a free-body diagram for the same.
Now, write the equation of force.
F=-T-mg
The acceleration of the object is in the negative y-axis plane, hence
-ma=-T-mg
Therefore, the tension on the spring is equal to
T=ma-mg
T=m(a-g)
Now, substitute the given values
T=0.2kg*(1-9.8)m/s2
T=0.2kg*(-8.8)m/s2
T=-1.76N
The tension on the sprig is -1.76N.
Read more on How To Find Normal Force With Tension: Several Approaches and Problem Examples.
Negative Tension Examples
There are various examples of the negative tension that we often come across. Let us ponder upon some examples.
Drowning of Bolt in the Water
Consider a bolt tied to a tread dropped in a glass of water. The molecular density of the bolt is more than the water, the bolt will immerse in the water accelerating down to the bottom of the glass.
If we write the equation of force for tension, then we have
F=-T-mg
T=-F-mg
The acceleration of the bolt is downward, hence,
T=-(-ma)-mg
T=m(a-g)
If the mass of the bolt is 4 grams, and the acceleration is 0.03m/s2, then tension on the tread is
T=4*10-3kg* (0.03-9.8)m/s2
T=4*10-3kg* (-9.77)m/s2
T=-0.039N
The tension on the tread is -0.039 Newton.
Lantern Hanging on the Hook with String Suddenly Detach and Falls Down
When the string detached from the lantern, the weight of the lantern is larger than the tension across the string, and hence the lantern accelerates downward.
Spring Shoes or Jumping Shoes
These shoes come with a spring attached beneath the shoes or a bouncer. When the body weight falls on the shoe it compresses. This time the body accelerates little downward, and the tension applied on the spring is also acting downward. Well, due to the potential energy accumulation in the spring and because of its elastic nature it regains its shape. That is why the spring is used in shoes to jump higher.
A Ladder on the Helicopter
Imagine that there is no person standing on the latter and the helicopter is accelerating in the upward direction. The tension may be applicable to the latter due to the air resistance. Well, the acceleration is upward and the tension is in the negative direction.
The mass m=0, hence the equation of force will be
F=-T
That is T=-ma.
Skipping
Tension on a massless rope is always zero. Because the tension on one end of the rope is canceled out by the equal and opposite tension from the other end. While skipping, the tension is exerted across the rope, but it is not positive because no pulling force is applied on the rope. The rope undergoes air resistance and the tension is created in the rope due to centrifugal force.
Loosening a Guitar strings
The tension on the strings on the guitar is created to generate the sound by bending the string or strumming it. If we loosen the string, then the tension on the string will be negative.
Balloon Floating in the Air
A balloon is filled with helium gas which is very light and hence freely floats to rise above the air. The density of helium is very less compared to the air molecules.
The balloon experiences a buoy force that carries it in the upward direction. The net force on the balloon is given by the equation
Fbuoy=ρ vg=-T-mg
Hence, tension is equal to
T=-ρvg-mg
T=-(ρ v+m)g
The tension on the tread of the balloon is negative.
Read more on Tension.
Frequently Asked Questions
Is stress a negative tension?
Stress is applied in such a way that it results in pulling two objects apart from each other, and it is called tensional stress.
Tensional stress is exerted in two opposite directions, separating or pulling the objects away from each other. If the tension is acting on the x-axis, then the tension in the left hand side is a negative tension.
Why compression is a negative tension?
The force applied to reduce the volume or size of the object is called compression.
On compression the tension is acting in the negative y-direction along with the weight of the object, then acceleration could be in positive or negative axis, the tension is always negative.
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Hi, I’m Akshita Mapari. I have done M.Sc. in Physics. I have worked on projects like Numerical modeling of winds and waves during cyclone, Physics of toys and mechanized thrill machines in amusement park based on Classical Mechanics. I have pursued a course on Arduino and have accomplished some mini projects on Arduino UNO. I always like to explore new zones in the field of science. I personally believe that learning is more enthusiastic when learnt with creativity. Apart from this, I like to read, travel, strumming on guitar, identifying rocks and strata, photography and playing chess.