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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.

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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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