Angular Momentum

Consider a point mass m moving with a velocity v,

Angular Momentum

It will have a momentum which can be given by mv, momentum describes how hard it is to stop an object in motion.

Consider the same point mass attached to a string, the string is tied to a point, and now if we exert a torque on the point mass, it will start rotating around the centre,

Torque

The particle of mass m will travel with a perpendicular velocity V which is the velocity that is perpendicular to the radius of the circle; r is the distance of the particle for the centre of its rotation.

The magnitude of \( \overrightarrow{L} \)   is given by

L = rmv sin \( \phi \)

  = r \( p_{\perp} \)

= rm\( v_{\perp} \)

= \( r_{\perp} p \)

  = \( r_{\perp} mv \)

Where Φ is the angle between \( \overrightarrow{r}\) and \( \overrightarrow{p}\) , \( p_{\perp}~ and~ v_{\perp} \)  are the components of \( \overrightarrow{p}\)  and \( \overrightarrow{v}\) perpendicular to \( \overrightarrow{r}\) . \( r_{\perp} \)  is the perpendicular distance between the fixed point and the extension of  \( \overrightarrow{p} \) .

Angular Momentum

The direction of angular momentum is given by the right-hand rule, which states that if you position your right hand such that the fingers are in the direction of \( \overrightarrow{r}\) . Then curl them around your palm such that they point towards the direction of \( \overrightarrow{p}\)  .The outstretched thumb gives the direction of \( \overrightarrow{l}\).

Notice the equation L = \( r_{\perp} mv \) the angular momentum of the body only changes when there is a net torque applied on it, so when there is no torque applied the perpendicular

Angular Momentum

velocity of the body will depend upon the radius of the circle, i.e. the distance from centre of mass of the body to the centre of the circle, therefore if the radius is reduced the velocity will increase and if the radius is increased the velocity will reduce, this is to conserve the angular momentum of the body.

When an ice-skater goes for a spin she starts off with her hands and leg far apart from the centre of her body, but when she needs more velocity to spin, she gets her hands and leg closer to her body, hence her angular momentum is conserved and she spins faster.

A gyroscope uses the principle of angular momentum to maintain its orientation, it utilises a spinning wheel which has 3 degrees of freedom, when it is rotated at high speed it locks on to the orientation, and it won’t deviate from its orientation, this is useful in space applications where the attitude of a spacecraft is really important factor to be controlled.To know more about angular momentum and its application contact our mentors at byjus.com.


Practise This Question

Two small balls A and B, each of mass m, are attached rigidly to the ends of a light rod of length d. The structure rotates about the perpendicular bisector of the rod at an angular speed ω. Calculate the angular momentum of the system about the axis of rotation