Two particle of masses $m_{1}$ and $m_{2}$ are joined by a light rigid rod of length $r$ . The system rotates at an angular speed $ω$ about an axis through the center of mass of the system and perpendicular to the rod. Show that the angular momentum of the system is $L = μ r^{2} ω$ where $μ$ is the reduced mass of the system defined as $μ = \frac{m_{1} m_{2}}{m_{1} + m_{2}}$

Open in App

Solution

The distance of the center of mass of the system from block of mass $m_{1}$ is:
$r_{1} = \frac{m_{2} r}{m_{1} + m_{2}}$
Angular momentum due to the mass $m_{1}$ at the centre of system is $= m_{1} r_{1}^{2} ω$
$= m_{1} {(\frac{m_{2} r}{m_{1} + m_{2}})}^{2} ω = \frac{m_{1} m_{2}^{2} r^{2}}{(m_{1} + m_{2})^{2}} ω$ ...... (1)

The distance of the center of mass of the system from block of mass $m_{2}$ is:
$r_{2} = \frac{m_{1} r}{m_{1} + m_{2}}$

Similarly the angular momentum due to the mass $m_{2}$ at the centre of system is $m_{2} r_{2}^{2} ω$
$= m_{2} {(\frac{m_{1} r}{m_{1} + m_{2}})}^{2} ω = \frac{m_{2} m_{1}^{2}}{(m_{1} + m_{2})^{2}} ω$ ......(2)

Therefore net angular momentum
$L_{n e t} = \frac{m_{1} m_{2}^{2} r^{2} ω}{(m_{1} + m_{2})^{2}} + \frac{m_{2} m_{1}^{2} r^{2} ω}{(m_{1} + m_{2})^{2}}$

$\Rightarrow \frac{m_{1} m_{2} (m_{1} + m_{2}) r^{2} ω}{(m_{1} + m_{2})^{2}} = \frac{m_{1} m_{2}}{(m_{1} + m_{2})} r^{2} ω = μ r^{2} ω$

Suggest Corrections

Similar questions

Q. Two particles of masses m₁ and m₂ are joined by a light rigid rod of length r. The system rotates at an angular speed ω about an axis through the centre of mass of the system and perpendicular to the rod. Show that the angular momentum of the system is

L = μ r^{2} ω

where

μ

is the reduced mass of the system defined as

μ = \frac{m_{1} + m_{2}}{m_{1} + m_{2}}

Two particles of masses $m_{1}$ and $m_{2}$ are joined by a light rigid rod of length r. The system rotates at an angular speed

$ω$ about an axis through the centre of mass of the system and perpendicular to the rod. Show that the angular momentum of the system is L = $μ r^{2} ω$ where $μ$ is the reduced mass of the system defined as $μ = \frac{m_{1} + m_{2}}{m_{1} + m_{2}}$

Q. Two balls of masses m and

2 m

are attached to the ends of a light rod of length L. The rod rotates with an angular speed

ω

about an axis passing through the centre of mass of system and perpendicular to the place. Find the angular momentum of the system about the axis of rotation.

Q. Two particles balls A and B, each of mass m, are attached rigidly to the ends of a light rod of length l. The system rotates about the perpendicular bisector of the rod at an angular speed

ω

. Calculate the angular momentum of the individual particles and of the system about the axis of rotation.

Q. Two point masses

m_{1}

and

m_{2}

are joined by a weightless rod of length

r

. Calculate the moment of inertia of the system about an axis passing through its centre of mass and perpendicular to the rod.

Join BYJU'S Learning Program