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An extended object which does not change shape. |
Motivation for Concept
Many everyday objects are essentially rigid bodies. Any object which does not significantly deform in its everyday use can be treated as a rigid body. Some examples are ceramic cups, keys, wooden chairs, and hockey pucks.
These objects should be contrasted with other objects that significantly deform when forces are applied. Examples are handbags, unrolled newspapers, cords, and beanbags.
Effects of External Forces on Rigid Bodies
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Pure Translational Acceleration
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Forces applied to an initially non-rotating rigid body in such a way that their line of action passes through the body's center of mass will produce pure translation of the rigid body. The acceleration of every point in the rigid body will be identical and governed by Newton's 2nd Law applied to the entire body:
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{excerpt}An extended object in which the distance between any two points in the object is constant in time.{excerpt} ||Page Contents|| |{toc}| ---- h2. Motivation of Concept Many everyday objects are essentially rigid bodies. Any object which does not significantly deform in its everyday use can be treated as a rigid body. Some examples are ceramic cups, keys, wooden chairs, and hockey pucks. These objects should be contrasted with other objects that significantly deform when forces are applied. Examples are handbags, unrolled newspapers, cords, and beanbags. ---- h2. Effects of External Forces on Rigid Bodies h4. Pure Translational Acceleration Forces applied to an initially non-rotating rigid body in such a way that their [line of action] passes through the body's center of mass will produce pure translation of the rigid body. The acceleration of every point in the rigid body will be identical and governed by [Newton's 2nd Law|Newton's Second Law] applied to the entire body: {latex}\begin{large}\[ \sum \vec{F}_{\rm ext} = M_{\rm tot}\vec{a}\] \end{large}{latex} where _M_~tot~ is the mass of the entire body. {table}{tr}{td}!davetranslate.png|width=50%!{td}{tr}{table} h4. Translational Acceleration of a Rotating Body If external forces are applied to a body that is initially rotating about its center of mass with their lines of action passing through the body's center of mass, then the acceleration of the center of mass will still be given by the formula: {latex} |
where Mtot is the mass of the entire body.
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Translational Acceleration of a Rotating Body
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If external forces are applied to a body that is initially rotating about its center of mass with their lines of action passing through the body's center of mass, then the acceleration of the center of mass will still be given by the formula:
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\begin{large}\[ \sum \vec{F}_{\rm ext} = M_{\rm tot}\vec{a}_{\rm cm}\] \end{large}{latex}
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The
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rest
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of
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the
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body
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will
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move
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along
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with
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the
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center
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of
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mass,
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but
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because
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of
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the
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rotation
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the
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acceleration
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of
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a
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given
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point
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p
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will
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differ
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from
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that
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of
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the
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cm.
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The
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acceleration
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of
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each
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point
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can
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be
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found
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by
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taking
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the
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vector
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sum
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of
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the
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centripetal
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acceleration
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needed
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to
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maintain
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the
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rotation.
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Pure Rotational Acceleration
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Even in the case that all external forces sum to zero, an extended rigid body may experience a change in its state of motion. If the torques resulting from the applied forces do not sum to zero, the rigid body will experience an angular acceleration about its center of mass, changing its rotational state. Assuming that the body possesses certain rotational symmetries, the angular acceleration of every point in the body will be identical and will be governed by the formula:
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h4. Pure Rotational Acceleration Even in the case that all external forces sum to zero, an extended rigid body may experience a change in its state of motion. If the [torques|torque (single-axis)] resulting from the applied forces do not sum to zero, the rigid body will experience an [angular acceleration] about its center of mass, changing its rotational state. Assuming that the body possesses certain [rotational symmetries|angular momentum about a single axis#rotational symmetries], the angular acceleration of every point in the body will be identical and will be governed by the formula: {latex}\begin{large}\[ \sum \tau = I\alpha \]\end{large}{latex} where _I_ is the [moment of inertia] of the body about the axis of the resulting rotation. {table}{tr}{td}!daverotate.png|width=50%!{td}{tr}{table} h4. Combined Rotational and Translational Acceleration If neither the forces nor the torques sum to zero, the motion of the rigid body can be treated as the sum of the resulting translational acceleration and rotational acceleration found by the methods described in the previous sections. Since the center of mass is the center of the rotation, its linear acceleration is the most easily obtained. It is given by: {latex} |
where I is the moment of inertia of the body about the axis of the resulting rotation.
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Combined Rotational and Translational Acceleration
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If neither the forces nor the torques sum to zero, the motion of the rigid body can be treated as the sum of the resulting translational acceleration and rotational acceleration found by the methods described in the previous sections. Since the center of mass is the center of the rotation, its linear acceleration is the most easily obtained. It is given by:
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\begin{large}\[ \sum \vec{F}_{\rm ext} = M_{\rm tot}\vec{a}_{\rm cm}\] \end{large}{latex}
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The
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angular
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acceleration
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of
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the
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entire
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body
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about
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the
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center
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of
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mass
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is:
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}\begin{large}\[ \sum \tau = I\alpha \]\end{large}{latex} {table}{tr}{td}!davetransrotate.png|width=50%!{td}{tr}{table} ---- h2. Relevant Models {children:page=Angular Momentum and Torque|all=true} ---- h2. Relevant Examples See any of the [Examples from Angular Momentum and Torque]. |
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