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  • Understand the relationship between force, mass and acceleration.
  • Study the turning effect of force.
  • Study the analogy between force and torque, mass and moment of inertia, and linear acceleration and angular acceleration.

If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity as seen in [link] . In fact, your intuition is reliable in predicting many of the factors that are involved. For example, we know that a door opens slowly if we push too close to its hinges. Furthermore, we know that the more massive the door, the more slowly it opens. The first example implies that the farther the force is applied from the pivot, the greater the angular acceleration; another implication is that angular acceleration is inversely proportional to mass. These relationships should seem very similar to the familiar relationships among force, mass, and acceleration embodied in Newton’s second law of motion. There are, in fact, precise rotational analogs to both force and mass.

The given figure shows a bike tire being pulled by a hand with a force F backward indicated by a red horizontal arrow that produces an angular acceleration alpha indicated by a curved yellow arrow in counter-clockwise direction.
Force is required to spin the bike wheel. The greater the force, the greater the angular acceleration produced. The more massive the wheel, the smaller the angular acceleration. If you push on a spoke closer to the axle, the angular acceleration will be smaller.

To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force F size 12{F} {} on a point mass m size 12{m} {} that is at a distance r size 12{r} {} from a pivot point, as shown in [link] . Because the force is perpendicular to r size 12{r} {} , an acceleration a = F m size 12{a= { {F} over {m} } } {} is obtained in the direction of F size 12{F} {} . We can rearrange this equation such that F = ma size 12{F= ital "ma"} {} and then look for ways to relate this expression to expressions for rotational quantities. We note that a = size 12{a=rα} {} , and we substitute this expression into F = ma size 12{F= ital "ma"} {} , yielding

F = mr α . size 12{F= ital "mr"α"."} {}

Recall that torque    is the turning effectiveness of a force. In this case, because F size 12{"F"} {} is perpendicular to r size 12{r} {} , torque is simply τ = Fr size 12{τ=rα} {} . So, if we multiply both sides of the equation above by r size 12{r} {} , we get torque on the left-hand side. That is,

rF = mr 2 α size 12{ ital "rF"= ital "mr" rSup { size 8{2} } α} {}

or

τ = mr 2 α. size 12{τ= ital "mr" rSup { size 8{2} } α.} {}

This last equation is the rotational analog of Newton’s second law ( F = ma size 12{F= ital "ma"} {} ), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and mr 2 size 12{ ital "mr" rSup { size 8{2} } } {} is analogous to mass (or inertia). The quantity mr 2 size 12{ ital "mr" rSup { size 8{2} } } {} is called the rotational inertia    or moment of inertia    of a point mass m size 12{m} {} a distance r size 12{r} {} from the center of rotation.

The given figure shows an object of mass m, kept on a horizontal frictionless table, attached to a pivot point, which is in the center of the table, by a cord that supplies centripetal force. A force F is applied to the object perpendicular to the radius r, which is indicated by a red arrow tangential to the circle, causing the object to move in counterclockwise direcion.
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 size 12{F} {} is applied to the object perpendicular to the radius r size 12{r} {} , causing it to accelerate about the pivot point. The force is kept perpendicular to r size 12{r} {} .

Making connections: rotational motion dynamics

Dynamics for rotational motion is completely analogous to linear or translational dynamics. Dynamics is concerned with force and mass and their effects on motion. For rotational motion, we will find direct analogs to force and mass that behave just as we would expect from our earlier experiences.

Questions & Answers

A stone is dropped down a well, if it take 5 seconds to reach the water, how dip is the well
Mollamin Reply
an aircraft at as steady velocity of 70m/so eastwards at a height of 800me drops a package of supplies .a, how long will it take for the package to rich the ground? b, how fast will it be going as it lands?
Ng Reply
what is hypothesis theory law
Tamba
physics is the science of measurement
Jide Reply
What is physics
Victor Reply
what is physics
Obaro
Good question! Physics is the study of the nature world . Does this help?
Yonn
physics is the study of matter in relations to energy.
Enoch
physics is the science of measurements
Jide
physics is a science concern with nature and properties of matter and energy
Ugomma
what is a parallelogram law of motion?
Nancy
Definition for physics
Adesola Reply
It deal with matter and relation to energy
Soughie
physics is the Study of matter in relation to energy.
albert
physics is a natural science that study matter its behaviour and relation to energy.
mohammed
physic tells us more about quantities and measurement also
Kelly
life as we know it that can be measured and calculated
Jesus
what is a reference frame
Chukwu Reply
what is anatomy in relation to physics
Mubarak Reply
how does half life exist
Humble Reply
 The amount of time it takes a radioactive isotope to decay into a stable isotope is different for each radioactive isotope, and is characterized by its “half-life”. An isotope's half-life is the amount of time it takes for half the number of atoms of that isotope to decay to another isotope.
Nardine
what is the difference between Mass and weight
Pjustin
mass is constant while weight varies. unit of mass is kg, unit of weight is newton
Faith
how can a coin float in water and what principle governs it
Mercy Reply
in my opinion that work of surface tension but restrictions on coin is that coin do not break surface energy of molecules but some days before scientists prove that's another types of force
Aman
which force hold floating coins together thats my confusion
Aman
how can a coin float in water and what principle governs it
Mercy Reply
why many of the coin floating in water
Aman Reply
what is hook's law
Momoh Reply
provided the elastic limit is not exceeded,the extension produce in an elastic material is equal to the applied force
Mahmud
What is Andromeda
Chinecherem Reply
👡
Mahmud
What are the international agreed system of unit for physical measurements
Aisher Reply
m, kg, s
Faith
kg , m, s
Faith
si
Aman
meter , kilogram , second
Kelly
international system of units is agreed system because more units are not in mks system so si exist with ampere unit
Aman
what is thermodynasmics
Femkid Reply
what is electromagnetic force
Femkid
Electromagnetic force means a force that deals with electric and magnetic
Nebil
Practice Key Terms 3

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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