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Calculating torque on a rigid body

[link] shows several forces acting at different locations and angles on a flywheel. We have | F 1 | = 20 N , | F 2 | = 30 N , | F 3 | = 30 N , and r = 0.5 m . Find the net torque on the flywheel about an axis through the center.

Figure shows a flywheel with three forces acting on it at different locations and angles. Force F3 is applied at the center and is perpendicular to the axis of rotation. Force F2 is applied at the left edge and is perpendicular to the axis of rotation. Force F1 is applied at the center and forms 30 degree angle with the axis of rotation.
Three forces acting on a flywheel.

Strategy

We calculate each torque individually, using the cross product, and determine the sign of the torque. Then we sum the torques to find the net torque.

Solution

We start with F 1 . If we look at [link] , we see that F 1 makes an angle of 90 ° + 60 ° with the radius vector r . Taking the cross product, we see that it is out of the page and so is positive. We also see this from calculating its magnitude:

| τ 1 | = r F 1 sin 150 ° = 0.5 m ( 20 N ) ( 0.5 ) = 5.0 N · m .

Next we look at F 2 . The angle between F 2 and r is 90 ° and the cross product is into the page so the torque is negative. Its value is

| τ 2 | = r F 2 sin 90 ° = −0.5 m ( 30 N ) = −15.0 N · m .

When we evaluate the torque due to F 3 , we see that the angle it makes with r is zero so r × F 3 = 0 . Therefore, F 3 does not produce any torque on the flywheel.

We evaluate the sum of the torques:

τ net = i | τ i | = 5 15 = −10 N · m .

Significance

The axis of rotation is at the center of mass of the flywheel. Since the flywheel is on a fixed axis, it is not free to translate. If it were on a frictionless surface and not fixed in place, F 3 would cause the flywheel to translate, as well as F 1 . Its motion would be a combination of translation and rotation.

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Check Your Understanding A large ocean-going ship runs aground near the coastline, similar to the fate of the Costa Concordia , and lies at an angle as shown below. Salvage crews must apply a torque to right the ship in order to float the vessel for transport. A force of 5.0 × 10 5 N acting at point A must be applied to right the ship. What is the torque about the point of contact of the ship with the ground ( [link] )?

Figure shows a ship that lies at an angle on the seashore. A force of 50000 N is applied at 10 degree angle to the normal at a point that 100 meters above the point of contact between the ship and the seashore.
A ship runs aground and tilts, requiring torque to be applied to return the vessel to an upright position.

The angle between the lever arm and the force vector is 80 ° ; therefore, r = 100 m(sin80 ° ) = 98.5 m .

The cross product τ = r × F gives a negative or clockwise torque.

The torque is then τ = r F = −98.5 m ( 5.0 × 10 5 N ) = −4.9 × 10 7 N · m .

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Summary

  • The magnitude of a torque about a fixed axis is calculated by finding the lever arm to the point where the force is applied and using the relation | τ | = r F , where r is the perpendicular distance from the axis to the line upon which the force vector lies.
  • The sign of the torque is found using the right hand rule. If the page is the plane containing r and F , then r × F is out of the page for positive torques and into the page for negative torques.
  • The net torque can be found from summing the individual torques about a given axis.

Conceptual questions

What three factors affect the torque created by a force relative to a specific pivot point?

magnitude of the force, length of the lever arm, and angle of the lever arm and force vector

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Give an example in which a small force exerts a large torque. Give another example in which a large force exerts a small torque.

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When reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?

The moment of inertia of the wheels is reduced, so a smaller torque is needed to accelerate them.

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Questions & Answers

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what are the possible sources of error in coefficient of static and dynamic friction and there precautions
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what is Bohr
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He is a physicist who formulated the atomic model of an Atom
Lily
And made 3 postulates
Lily
Check university physics vol 3 > Nuclear physics
Lily
Bohr model
kami
what is mean by Doppler effect
RAMANJI Reply
Good
Ahmad
increase or decrease in the frequency of sound and light.
Jhon
good
Eng
is it?
Vinayaka
actually it is apparent change in the frequency of light or sound as object move towards or away.
Vinayaka
state the basic assumption of kinetic theory of gases
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state the characteristics of gases that differentiate them from solids
FELIX
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formula for Velocity
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Elijah Reply
what is the formula for frictional force
bassey Reply
I believe, correct me if I am wrong, but Ffr=Fn*mu
Grant
frictional force ,mathematically Fforce (Ffr) =K∆R where by K stands for coefficient of friction ,R stands for normal force/reaction NB: R = mass of a body ( m) x Acc.due gravity (g) The formula will hold the meaning if and only if the body is relatively moving with zero angle (∅ = 0°C)
Boay
What is concept associated with linear motion
Adeoye Reply
what causes friction?
Elijah
uneven surfaces cause friction Elijah
Shii
rough surfacea
Grant
what will happen to vapor pressure when you add solute to a solution?
shane Reply
how is freezing point depression different from boiling point elevation?
shane
how is the osmotic pressure affect the blood serum?
shane
what is the example of colligative properties that seen in everyday living?
shane
freezing point depression deals with the particles in the matter(liquid) loosing energy.....while boiling elevation is the particles of the matter(liquid)gaining energy
E-vibes
What is motion
Adeoye Reply
moving place to place
Addis
change position with respect to surrounding
Muhammad
to which
Addis
to where ?
Addis
the phenomenon of an object to changes its position with respect to the reference point with passage of time then it is called as motion
Shubham
it's just a change in position
festus
reference point -it is a fixed point respect to which can say that a object is at rest or motion
Shubham
yes
Shubham
A change in position
Lily
change in position depending on time
bassey
a change in the position of a body
E-vibes
Is there any calculation for line integral in scalar feild?
Sadia Reply
what is thrust
Aarti Reply
when an object is immersed in liquid, it experiences an upward force which is called as upthrust.
Phanindra
@Phanindra Thapa No, that is buoyancy that you're talking about...
Shii
thrust is simply a push
Shii
it is a force that is exerted by liquid.
Phanindra
what is the difference between upthrust and buoyancy?
misbah
The force exerted by a liquid is called buoyancy. not thrust. there are many different types of thrust and I think you should Google it instead of asking here.
Sharath
hey Kumar, don't discourage somebody like that. I think this conversation is all about discussion...remember that the more we discuss the more we know...
festus
thrust is an upward force acting on an object immersed in a liquid.
festus
uptrust and buoyancy are the same
akanbi
the question isn't asking about up thrust. he simply asked what is thrust
Shii
a Thrust is simply a push
Shii
the perpendicular force applied on the body
Shubham
thrust is a force of depression while
bassey
what is friction?
MFON
while upthrust is a force that act on a body when it is fully or partially submerged in a liquid
bassey
mathematically upthrust (u) = Real weight (wr) - Apparent weight (wa) u = wr- wa.
Boay
friction is a force which opposes relative motion.
Boay
Practice Key Terms 2

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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