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Angular acceleration
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<h2 id="orbital-angular-acceleration-of-a-point-particle">Orbital angular acceleration of a point particle</h2> <h3>Particle in two dimensions</h3> <p>In two dimensions, the orbital angular acceleration is the rate at which the two-dimensional orbital angular velocity of the particle about the origin changes. The instantaneous angular velocity <i>ω</i> at any point in time is given by </p> ω = v ⊥ r , {\displaystyle \omega ={\frac {v_{\perp }}{r}},} <p>where r {\displaystyle r} is the distance from the origin and v ⊥ {\displaystyle v_{\perp }} is the cross-radial component of the instantaneous velocity (i.e. the component perpendicular to the position vector), which by convention is positive for counter-clockwise motion and negative for clockwise motion. </p><p>Therefore, the instantaneous angular acceleration <i>α</i> of the particle is given by<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> α = d d t ( v ⊥ r ) . {\displaystyle \alpha ={\frac {d}{dt}}\left({\frac {v_{\perp }}{r}}\right).} <p>Expanding the right-hand-side using the product rule from differential calculus, this becomes </p> α = 1 r d v ⊥ d t − v ⊥ r 2 d r d t . {\displaystyle \alpha ={\frac {1}{r}}{\frac {dv_{\perp }}{dt}}-{\frac {v_{\perp }}{r^{2}}}{\frac {dr}{dt}}.} <p>In the special case where the particle undergoes circular motion about the origin, d v ⊥ d t {\displaystyle {\frac {dv_{\perp }}{dt}}} becomes just the tangential acceleration a ⊥ {\displaystyle a_{\perp }} , and d r d t {\displaystyle {\frac {dr}{dt}}} vanishes (since the distance from the origin stays constant), so the above equation simplifies to </p> α = a ⊥ r . {\displaystyle \alpha ={\frac {a_{\perp }}{r}}.} <p>In two dimensions, angular acceleration is a number with plus or minus sign indicating orientation, but not pointing in a direction. The sign is conventionally taken to be positive if the angular speed increases in the counter-clockwise direction or decreases in the clockwise direction, and the sign is taken negative if the angular speed increases in the clockwise direction or decreases in the counter-clockwise direction. Angular acceleration then may be termed a <a href="/facts/Pseudoscalar/3ahEmnIW">pseudoscalar</a>, a numerical quantity which changes sign under a <a href="/facts/Parity_(physics)/emSbJ4hD">parity inversion</a>, such as inverting one axis or switching the two axes. </p> <h3>Particle in three dimensions</h3> <p>In three dimensions, the orbital angular acceleration is the rate at which three-dimensional orbital angular velocity vector changes with time. The instantaneous angular velocity vector ω {\displaystyle {\boldsymbol {\omega }}} at any point in time is given by </p> ω = r × v r 2 , {\displaystyle {\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}},} <p>where r {\displaystyle \mathbf {r} } is the particle's position vector, r {\displaystyle r} its distance from the origin, and v {\displaystyle \mathbf {v} } its velocity vector.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>Therefore, the orbital angular acceleration is the vector α {\displaystyle {\boldsymbol {\alpha }}} defined by </p> α = d d t ( r × v r 2 ) . {\displaystyle {\boldsymbol {\alpha }}={\frac {d}{dt}}\left({\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}\right).} <p>Expanding this derivative using the product rule for cross-products and the ordinary quotient rule, one gets: </p> α = 1 r 2 ( r × d v d t + d r d t × v ) − 2 r 3 d r d t ( r × v ) = 1 r 2 ( r × a + v × v ) − 2 r 3 d r d t ( r × v ) = r × a r 2 − 2 r 3 d r d t ( r × v ) . {\displaystyle {\begin{aligned}{\boldsymbol {\alpha }}&={\frac {1}{r^{2}}}\left(\mathbf {r} \times {\frac {d\mathbf {v} }{dt}}+{\frac {d\mathbf {r} }{dt}}\times \mathbf {v} \right)-{\frac {2}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {v} \right)\\\\&={\frac {1}{r^{2}}}\left(\mathbf {r} \times \mathbf {a} +\mathbf {v} \times \mathbf {v} \right)-{\frac {2}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {v} \right)\\\\&={\frac {\mathbf {r} \times \mathbf {a} }{r^{2}}}-{\frac {2}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {v} \right).\end{aligned}}} <p>Since r × v {\displaystyle \mathbf {r} \times \mathbf {v} } is just r 2 ω {\displaystyle r^{2}{\boldsymbol {\omega }}} , the second term may be rewritten as − 2 r d r d t ω {\displaystyle -{\frac {2}{r}}{\frac {dr}{dt}}{\boldsymbol {\omega }}} . In the case where the distance r {\displaystyle r} of the particle from the origin does not change with time (which includes circular motion as a subcase), the second term vanishes and the above formula simplifies to </p> α = r × a r 2 . {\displaystyle {\boldsymbol {\alpha }}={\frac {\mathbf {r} \times \mathbf {a} }{r^{2}}}.} <p>From the above equation, one can recover the cross-radial acceleration in this special case as: </p> a ⊥ = α × r . {\displaystyle \mathbf {a} _{\perp }={\boldsymbol {\alpha }}\times \mathbf {r} .} <p>Unlike in two dimensions, the angular acceleration in three dimensions need not be associated with a change in the angular <i>speed ω = | ω | {\displaystyle \omega =|{\boldsymbol {\omega }}|} </i>: If the particle's position vector "twists" in space, changing its instantaneous plane of angular displacement, the change in the <i>direction</i> of the angular velocity ω {\displaystyle {\boldsymbol {\omega }}} will still produce a nonzero angular acceleration. This cannot not happen if the position vector is restricted to a fixed plane, in which case ω {\displaystyle {\boldsymbol {\omega }}} has a fixed direction perpendicular to the plane. </p><p>The angular acceleration vector is more properly called a <a href="/facts/Pseudovector/c3b5GgWD">pseudovector</a>: It has three components which transform under rotations in the same way as the Cartesian coordinates of a point do, but which do not transform like Cartesian coordinates under reflections. </p> <h3>Relation to torque</h3> <p>The net <i><a href="/facts/Torque/KcrcvNtV">torque</a></i> on a point particle is defined to be the pseudovector </p> τ = r × F , {\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} ,} <p>where F {\displaystyle \mathbf {F} } is the net force on the particle.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>Torque is the rotational analogue of force: it induces change in the rotational state of a system, just as force induces change in the translational state of a system. As force on a particle is connected to acceleration by the equation F = m a {\displaystyle \mathbf {F} =m\mathbf {a} } , one may write a similar equation connecting torque on a particle to angular acceleration, though this relation is necessarily more complicated.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>First, substituting F = m a {\displaystyle \mathbf {F} =m\mathbf {a} } into the above equation for torque, one gets </p> τ = m ( r × a ) = m r 2 ( r × a r 2 ) . {\displaystyle {\boldsymbol {\tau }}=m\left(\mathbf {r} \times \mathbf {a} \right)=mr^{2}\left({\frac {\mathbf {r} \times \mathbf {a} }{r^{2}}}\right).} <p>From the previous section: </p> α = r × a r 2 − 2 r d r d t ω , {\displaystyle {\boldsymbol {\alpha }}={\frac {\mathbf {r} \times \mathbf {a} }{r^{2}}}-{\frac {2}{r}}{\frac {dr}{dt}}{\boldsymbol {\omega }},} <p>where α {\displaystyle {\boldsymbol {\alpha }}} is orbital angular acceleration and ω {\displaystyle {\boldsymbol {\omega }}} is orbital angular velocity. Therefore: </p> τ = m r 2 ( α + 2 r d r d t ω ) = m r 2 α + 2 m r d r d t ω . {\displaystyle {\boldsymbol {\tau }}=mr^{2}\left({\boldsymbol {\alpha }}+{\frac {2}{r}}{\frac {dr}{dt}}{\boldsymbol {\omega }}\right)=mr^{2}{\boldsymbol {\alpha }}+2mr{\frac {dr}{dt}}{\boldsymbol {\omega }}.} <p>In the special case of constant distance r {\displaystyle r} of the particle from the origin ( d r d t = 0 {\displaystyle {\tfrac {dr}{dt}}=0} ), the second term in the above equation vanishes and the above equation simplifies to </p> τ = m r 2 α , {\displaystyle {\boldsymbol {\tau }}=mr^{2}{\boldsymbol {\alpha }},} <p>which can be interpreted as a "rotational analogue" to F = m a {\displaystyle \mathbf {F} =m\mathbf {a} } , where the quantity m r 2 {\displaystyle mr^{2}} (known as the <a href="/facts/Moment_of_inertia/TpLLAVl7">moment of inertia</a> of the particle) plays the role of the mass m {\displaystyle m} . However, unlike F = m a {\displaystyle \mathbf {F} =m\mathbf {a} } , this equation does <i>not</i> apply to an arbitrary trajectory, only to a trajectory contained within a spherical shell about the origin. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Angular_momentum/7TuSVFxq">Angular momentum</a></li> <li><a href="/facts/Angular_frequency/VqLm3L3R">Angular frequency</a></li> <li><a href="/facts/Angular_velocity/2n8JljAT">Angular velocity</a></li> <li><a href="/facts/Chirpyness/hCQrvcz8">Chirpyness</a></li> <li><a href="/facts/Rotational_acceleration/jDv9uzAS">Rotational acceleration</a></li> <li><a href="/facts/Torque/KcrcvNtV">Torque</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Rotational Variables". LibreTexts. MindTouch. 18 October 2016. Retrieved 1 July 2020. <a href="https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_I_-_Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves_(OpenStax)/10%3A_Fixed-Axis_Rotation__Introduction/10.02%3A_Rotational_Variables" target="_blank">https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_I_-_Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves_(OpenStax)/10%3A_Fixed-Axis_Rotation__Introduction/10.02%3A_Rotational_Variables</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Singh, Sunil K. Angular Velocity. Rice University. <a href="https://cnx.org/contents/MymQBhVV@175.14:51fg7QFb@14/Angular-velocity" target="_blank">https://cnx.org/contents/MymQBhVV@175.14:51fg7QFb@14/Angular-velocity</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Singh, Sunil K. Angular Velocity. Rice University. <a href="https://cnx.org/contents/MymQBhVV@175.14:51fg7QFb@14/Angular-velocity" target="_blank">https://cnx.org/contents/MymQBhVV@175.14:51fg7QFb@14/Angular-velocity</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Singh, Sunil K. Torque. Rice University. <a href="https://cnx.org/contents/MymQBhVV@175.14:JOsDHAfQ@4/Torque" target="_blank">https://cnx.org/contents/MymQBhVV@175.14:JOsDHAfQ@4/Torque</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Mashood, K.K. Development and evaluation of a concept inventory in rotational kinematics (PDF). Tata Institute of Fundamental Research, Mumbai. pp. 52–54. <a href="http://www.hbcse.tifr.res.in/research-development/ph.d.-theses/thesis-mashoodkk.pdf" target="_blank">http://www.hbcse.tifr.res.in/research-development/ph.d.-theses/thesis-mashoodkk.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>