<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Calculate the torques on rotating systems about a fixed axis to find the angular acceleration
  • Explain how changes in the moment of inertia of a rotating system affect angular acceleration with a fixed applied torque

In this section, we put together all the pieces learned so far in this chapter to analyze the dynamics of rotating rigid bodies. We have analyzed motion with kinematics and rotational kinetic energy but have not yet connected these ideas with force and/or torque. In this section, we introduce the rotational equivalent to Newton’s second law of motion and apply it to rigid bodies with fixed-axis rotation.

Newton’s second law for rotation

We have thus far found many counterparts to the translational terms used throughout this text, most recently, torque, the rotational analog to force. This raises the question: Is there an analogous equation to Newton’s second law, Σ F = m a , which involves torque and rotational motion? To investigate this, we start with Newton’s second law for a single particle rotating around an axis and executing circular motion. Let’s exert a force F on a point mass m that is at a distance r from a pivot point ( [link] ). The particle is constrained to move in a circular path with fixed radius and the force is tangent to the circle. We apply Newton’s second law to determine the magnitude of the acceleration a = F / m in the direction of F . Recall that the magnitude of the tangential acceleration is proportional to the magnitude of the angular acceleration by a = r α . Substituting this expression into Newton’s second law, we obtain

F = m r α .
Figure shows a table with a frictionless tabletop. An object with the mass m is supported by a horizontal frictionless table and is attached to a pivot point by a cord with the length r. A force F is applied to the object perpendicular to the cord r.
An object is supported by a horizontal frictionless table and is attached to a pivot point by a cord that supplies centripetal force. A force F is applied to the object perpendicular to the radius r , causing it to accelerate about the pivot point. The force is perpendicular to r .

Multiply both sides of this equation by r ,

r F = m r 2 α .

Note that the left side of this equation is the torque about the axis of rotation, where r is the lever arm and F is the force, perpendicular to r . Recall that the moment of inertia for a point particle is I = m r 2 . The torque applied perpendicularly to the point mass in [link] is therefore

τ = I α .

The torque on the particle is equal to the moment of inertia about the rotation axis times the angular acceleration . We can generalize this equation to a rigid body rotating about a fixed axis.

Newton’s second law for rotation

If more than one torque acts on a rigid body about a fixed axis, then the sum of the torques equals the moment of inertia times the angular acceleration:

i τ i = I α .

The term I α is a scalar quantity and can be positive or negative (counterclockwise or clockwise) depending upon the sign of the net torque. Remember the convention that counterclockwise angular acceleration is positive. Thus, if a rigid body is rotating clockwise and experiences a positive torque (counterclockwise), the angular acceleration is positive.

[link] is Newton’s second law for rotation    and tells us how to relate torque, moment of inertia, and rotational kinematics. This is called the equation for rotational dynamics    . With this equation, we can solve a whole class of problems involving force and rotation. It makes sense that the relationship for how much force it takes to rotate a body would include the moment of inertia, since that is the quantity that tells us how easy or hard it is to change the rotational motion of an object.

Questions & Answers

who discover periodic table?
lovet Reply
it wasn't discovered , it was made.and the person who made it was dmitri mendeleev. dobreinier and newland gave their laws before dmitri related periodic table but wasn't successful in their work
Nope, numerous number of scientist had actually contributed in the making of periodic table. Dmitri Mendeleev succeeded making all the elements into the right order in accordance to their atomic number.
what is the Greek name for calcium
Oniyide Reply
different types of wave
Yog Reply
longitudinal and transverse waves
a gun is kept in the state that it cannot move anywhere and the bullet is fired. Then what is the effect on the velocity of bullet and KE of gun ?
Gobinda Reply
want is meant by the term solar system
jafar Reply
it refers to the sun and all heavenly bodies revolving around it.
excatly...for sure
in addition to Danie's, a solar system is a collection of planets and their moons, asteroids, and other objects bound together by the Star's gravitational force directly or indirectly.
what is meant by total internal reflection
Akshay Reply
what iw meant by total internal reflection
Lorentz force?
study fibre optics. .you will get total internal reflection
what is Lorentz force?
a ray of light traveling at an angle of incidence greater than critical angle from denser to rarer medium is totally reflected back into the denser medium is called total internal reflaction
motion in strat line where is this chapter
Vijaybhai Reply
motion in straight line is kinematic's part
this defination isn't correct
motion in one dimension
what is Lorentz force?
jyotirmayee Reply
what is maxwell electromagnetic law?
at what angle should the two forces 2p and root 2p acts so that the resultant force is p root 10
Akshay Reply
what answer fir this
what's the working difference between a dynamo and a pump?
Piyali Reply
a dynamo is basically a dc generator while pump is usually equipped with a motor
a dynamo converts mechanical energy to electrical while a pump is opposite to that
why sea water looks bluish?
Piyali Reply
cuz the sky is blue...
you see the reflexion of the "blue" sky in the water
somewhere sea water turns green why?
never seen bro... are u sure ?
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
waaaww... you re awesome
because of the reflection of the sky
rays coming from the sun consist of all 7 colours ie.VIBGYOR. when the ray strikes surface of water,all colors gets absorbed by it except blue which gets reflected by it.so we find the sea water appearing bluish
how can someone identify sea water from rain water
i think it's not possible as because rainwater consists of water from all kind of water bodies ie.lakes,seas etc but u can predict if u have a sea nearby ur home or city
i need solutions of unuversity pbysucs volume 1
Vimla Reply
me too
Help us if anyone knows
bring questions
actually if u wanted whole book solution then u should buy the solution book
from where
where do u live ,,,,,if u live in Delhi then at bellsarayeiii or from stationary store ,,,,chatarpurrr ,,,,,there is a popular books store,,,,,u have to buy from there
Kota Rajasthan
so strange,,,,,r u preparing for pmt,,,?
for IIT
then concern near book store
in Rajasthan
or ask questions here
thanks for your help
its fine,,most welcome
what is three dimensional coordinate system?
Sachindra Reply
considering the change of vectors in all three dimensions of space
direction co-sign of vector questions
Vipin Reply
opposite mean
what is the different between action and reaction?
Asali Reply
action is external force. Reaction exists because of the external action. Reaction is mostly an internal force
The difference is said in the word itself. There is no existance of reaction without an application of action. So action occur first then reaction. Reaction may produce in body itself or to other body i.e., it may be internal or external.
or we can say , reaction is the result of action
Practice Key Terms 2

Get the best University physics vol... course in your pocket!

Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?