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05 - Circular Motion

Circular Motion

Circular Motion

Motion of bodies in circular path is called circular motion

Angular Motion

Circular motion of a body may also be called angular motion.

Angular Displacement

It is the angle swept by the radial line during circular motion of a particle measured from some initial point to some final point."

  • Angular displacement has direction along axis of rotation and can be determined by right hand rule
  • SI unit of angular displacement is radian
  • One radian is an angle made by an arc at the center, whose length is equal to the radius of circle
  • Definition of radian gives following useful relation: $$ s=r\theta $$
\[ 1^o=\frac{\pi}{180}rad= 0.0174 rad, \;1 rad = 57.3^o \]
  • Angular displacement is angle in radian or degree covered by body having circular motion
  • Non SI units are also used which are "degree" and "rev".

Do you know?

For very small values of angular displacement it is treated as vector quantity.

Angular Velocity & Acceleration

  • Rate of change of angular displacement is called angular velocity; usually not a vector quantity \(w_ang=\lim_{\Delta t\to0} \frac{\Delta\theta}{\Delta t}\) always vector quantity.

  • Tangential and angular velocities are related as: \(v=wr\) or \(\bar{v}=w*r\)

  • S.I unit of angular velocity is rads\(^{-1}\),
  • Rate of change of angular velocity is called angular acceleration, $$ \vec{\alpha} = \frac{\vec{\Delta \omega}}{\Delta t } $$
  • SI unit of angular acceleration is rads\(^{-2}\)
  • Angular acceleration is a vector Quantity related to tangential acceleration \(\vec{a}\) by the following formula \(a=r\alpha\)
  • Direction of angular acceleration and angular velocity is along axis of rotation,

Do you know?

The angular velocity is practically measured in revolution per minute (rpm) as they provide an easy interpretation of angular motion,

  • If angular velocity increases then \(\vec{\omega}\) and \(\alpha\) are in same direction and if angular velocity decreases , then \(\vec{\omega}\) and \(\vec{\alpha}\) are in opposite direction

Do you know?

All the points on a rigid body rotating about a fixed axis have same angular displacement, velocity and acceleration.

Rigid Body

A body, which maintains a constant distance between its two consecutive particles, when a definite load is applied to it.

Note

All the solid bodies can be treated as rigid in a specific range of loads only (eg wall is treated to be rigid for a human being but not for hummer.)

Centripetal Force & Acceleration

The force required to bend a straight line path of a body into the circular path is called centripetal force.

If the centripetal force is removed from the rotating object it will follow a straight' line motion confined on the tangent to that circular path.

Assumptions

To derive the equations for the centripetal acceleration we assume that speed of the object is constant so that the tangential component of velocity does not produce acceleration but radial component only. The equation is

\[ \bar{a}=\frac{\Delta\bar{v}}{\Delta t}\hat{a}_r+ \frac{v^2}{r}\hat{a}_r \]
  • In vector form, centripetal force and acceleration can be written as; $$ \vec{F_c}=-mr\omega ^2\hat{r}=-m\vec{r}\omega=-\left(\frac{mv ^2}{r}\right)\hat{r}=-\left(\frac{mv ^2}{r ^2}\right)\vec{r} $$

$$ \vec{a_c}=-r\omega 2\hat{r} = \bar{r}\omega=-\left(\frac{v2}{r})\hat{r}= -\left(\frac{v2}{r2}\right)\vec{r}$
* Work done by centripetal force is zero. * Centripetal and centrifugal forces form true action & reaction pair but they can't balance each other because they don't act on same body.

Comparision of Linear Motion & Angular Motion

Linear Motion Angular Motion
Linear displacement , \(\vec{d}\) Angular displacement, \(\vec{\theta}\)
Linear velocity, \(\bar{v}=\frac{\Delta d}{\Delta t}\) Angular velocity, \(\omega =\frac{\Delta\vec{\theta}}{\Delta t}\)
Acceleration or linear acceleration $$ \bar{a}=\frac{\Delta v}{\Delta t} $$ Angular acceleration,\(\vec{\alpha}=\frac{\Delta \omega}{\Delta t}\)
Mass, \(m\) Moment of inertia, \(I=mr^2\)
Linear momentum, \(\vec{P} = m\vec{v}\) Angular momentum, \(\vec{L}=I\bar{\omega}\)
Impulse, \(I\) or \(J=\vec{F}\times\Delta t\) Angular impulse \(=\vec{\tau}\times \Delta t\)
Force, \(\vec{F}=m\vec{a}=\frac{\Delta \bar{p} }{\Delta t}\) Torque, \(\vec{\tau}=I\vec{\alpha}=\frac{\Delta \vec{L}}{\Delta t}\)
Work, \(W=\bar{F}\bar{d}=\frac{1}{2}m(v ^2_2 - v ^2_2)\) Rotational work, $$ W=\tau\theta=\frac{1}{2} I(w ^2_2 -w ^2_2) $$
Kinetic energy, \(KE=\frac{1}{2}mv ^2\) Kinectic energy pf rotation \(KE=\frac{1}{2}I\omega^2\)
Newton's laws in linear motion:
  • First law: If \(F=0\) then \(v =\) constant
  • Second Law: \(\bar{F}=m\bar{a}\)
  • Third Law: \(\bar{F_{12}} = -\bar{F_{21}}\)
 Newton's laws in rotational motion:
  • First law: If \(\tau=0\) then \(\omega=\) constant
  • Second Law: \(\tau=I\alpha\)
  • Third Law: \(\bar{\tau_{12}} = -\bar{\tau_{21}}\)
Equations of linear motion
  • \(v_f=v_i+at\)
  • \(S=v_it+\frac{1}{2}at^2\)
  • \(v^2_f - v^2_i= 2aS\)
 Equations of linear motion
  • \(\omega_f=\omega_i+at\)
  • \(\theta=\omega_i t+\frac{1}{2}\alpha t^2\)
  • \(\omega^2_f - \omega^2_i= 2\alpha \theta\)
Distance covered in nth second $$ S_n=v_i+\frac{1}{2} a(2n-1) $$ Angle subtended in nth second $$ \theta_n=\omega_i+\frac{\alpha}{2}(2n-1) $$
$$ S=vt $$ $$ \theta=\omega t $$
$$ \lt v\gt=\frac{v_f+v_i}{2} $$ $$ \lt\omega\gt=\frac{\omega_f+\omega_i}{2} $$

Moment of inertia

The measure hindrance affered by a rigif body against angular motions, where \(\tau\) distuributing torque acts over tge body.

  • Mathematically expressed as moment of inertia \(=I=mr^2\).
  • It is measured in \(kgm^2\)
  • Moment of inertia plays same role in angular motion as the mass in linear motion.
  • The equation \(\tau=I\alpha\) is the rotational analogue of the Newton's second law \(F=ma\)

Moment of inertia depends upon

  1. Mass of body
  2. Distribution of the mass about the axis of rotation

Angular Momentum

  • Angular momentum is due to spin motion or orbital motion of a body, and is also called moment of momentum,

To be completed...

  • ANGULAR MOMENTUM OF A PARTICLE The angular momuntum of a particle of mass m with respect to a chosen axis is given by When it ,:ioser incresset weed sa.porrity Circular Motion my = p

The angular momentum of a particle of mass m with respect to a chosen axis is given by L = mvr sin 0 or more formally by the vector product
L=rxp The direction is given by the right hand rule which would give L the direction out of the diagram. For a circular orbit, L becomes L = rnvr 11W Angular Momentum The angular momentum of a rigid object is defined as the product of the moment of inertia and the angular velocity. itmoen nf iotat,on of the wha71 )Momentum- n,. Angola, MuiI Inertia x Voincity L = I x (,a L = co The nqht and rule for angular Angular momcnturn veCtOr. • Orbital angular momentum is greater than spin angular momentum of an electron. LAI OF CONSERVATION OF ANGULAR MOMENTUM Angular momentum is a conserved quantity only in isolated system. 1.; L L -constant if and only if E Tv, = OR CO 1 =126)2 • Total angular momentum of a particle or a system of particle remains constant Provided no net external torque acts on it. ticis When a umnast closes her arms while standing on a rotating joy wheel, the wheel speeds up, and slows down for the reverse just to conserve angular momentum. SPnrigboard diver has more rotation when she pulls her body into closed tuck Position, • Balance of a sport bicycle is maintained due to the conservation of angular 'onieritum for its thin rotating wheels,

To be completed...

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