Question
Obtain an expression for the angular momentum of a body rotating with uniform angular velocity.
✓
Answer
i. Consider a rigid object rotating with a constant angular speed ' $\omega$ ' about an axis perpendicular to the plane of the paper.
A body of $N$ particles
ii. Let us consider the object to be consisting of $N$ number of particles of masses $m_1, m_2, \ldots . . m_N$ at respective perpendicular distances $r_1, r_2, \ldots . . r_N$ from the axis of rotation.
iii. As the object rotates, all these particles perform UCM with the same angular speed $\omega$, but with different linear speeds $v_1=r_1 \omega$, $v_2=r_2 \omega, \ldots . . v_N=r_N \omega$.
Directions of individual velocities $\overrightarrow{ v }_1, \overrightarrow{ v }_2, \ldots \ldots \overrightarrow{ v }_{ N }$, are along the tangents to the irrespective tracks.
iv. Linear momentum of the first particle is of magnitude $p_1=m_1 v_1$ $=m_1 r_1 \omega$. Its direction is along that of $\vec{v}_1$.
Its angular momentum is thus of magnitude $L_1=p_1 r_1=m_1 r_1^2 \omega$ Similarly, $L _2= m _1 r _2^2 \omega, L _3= m _3 r _3^2 \omega, \ldots, L _{ N }= m _{ N } r _{ N }^2 \omega$.
$v$. For a rigid body with a fixed axis of rotation, all these angular momenta are directed along the axis of rotation, and this direction can be obtained by using the right-hand thumb rule. As all of them have the same direction, their magnitudes can be algebraically added.
vi. Thus, the magnitude of angular momentum of the body is given by
$ L = m _1 r _1^2 \omega+ m _1 r _2^2 \omega+\ldots .+ m _{ N } r _{ N }^2 \omega$
$=\left( m _1 r _1^2+ m _2 r _2^2+\ldots \ldots+ m _{ N } r _{ N }^2\right) \omega= I \omega $
where, $l = m _1 r _1^2+ m _2 r _2^2+\ldots .+ m _{ N } r _{ N }^2$ is the moment of inertia of the body about the given axis of rotation.
A body of $N$ particles
ii. Let us consider the object to be consisting of $N$ number of particles of masses $m_1, m_2, \ldots . . m_N$ at respective perpendicular distances $r_1, r_2, \ldots . . r_N$ from the axis of rotation.
iii. As the object rotates, all these particles perform UCM with the same angular speed $\omega$, but with different linear speeds $v_1=r_1 \omega$, $v_2=r_2 \omega, \ldots . . v_N=r_N \omega$.
Directions of individual velocities $\overrightarrow{ v }_1, \overrightarrow{ v }_2, \ldots \ldots \overrightarrow{ v }_{ N }$, are along the tangents to the irrespective tracks.
iv. Linear momentum of the first particle is of magnitude $p_1=m_1 v_1$ $=m_1 r_1 \omega$. Its direction is along that of $\vec{v}_1$.
Its angular momentum is thus of magnitude $L_1=p_1 r_1=m_1 r_1^2 \omega$ Similarly, $L _2= m _1 r _2^2 \omega, L _3= m _3 r _3^2 \omega, \ldots, L _{ N }= m _{ N } r _{ N }^2 \omega$.
$v$. For a rigid body with a fixed axis of rotation, all these angular momenta are directed along the axis of rotation, and this direction can be obtained by using the right-hand thumb rule. As all of them have the same direction, their magnitudes can be algebraically added.
vi. Thus, the magnitude of angular momentum of the body is given by
$ L = m _1 r _1^2 \omega+ m _1 r _2^2 \omega+\ldots .+ m _{ N } r _{ N }^2 \omega$
$=\left( m _1 r _1^2+ m _2 r _2^2+\ldots \ldots+ m _{ N } r _{ N }^2\right) \omega= I \omega $
where, $l = m _1 r _1^2+ m _2 r _2^2+\ldots .+ m _{ N } r _{ N }^2$ is the moment of inertia of the body about the given axis of rotation.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
What was Maxwell's concept of light?
A piece of straight wire has mass 20 g and length 1m. It is to be levitated using a current of 1 A flowing through it and a perpendicular magnetic field B in a horizontal direction. What must be the magnetic of B?
If a glass plate of refractive index $1.732$ is to be used as a polarizer, what would be the
$(i)$ polarizing angle and
$(ii)$ angle of refraction?
→$(i)$ polarizing angle and
$(ii)$ angle of refraction?
Write an expression for an alternating emf that varies sinusoidally with time. Show graphically variation of emf with time.
A solenoid $1.5 m$ long and $4 \ cm$ in diameter has $-10$ turns $/ \ cm$. A current of $5 A$ is passing through it. Calculate the magnetic induction $(i)$ inside $(ii)$ at one end on the axis of the solenoid.
→State the drawbacks of Newton's corpuscular theory of light.
Determine the change in wavelength of light during its passage from air to glass, if the refractive index of glass with respect to air is $1.5$ and the frequency of light is $5 \times 10^{14}\ Hz. [$Speed of light in air $=( c )=3 \times 10^8 m / s ]$
→In Young's experiment the ratio of intensity at the maxima and minima in the interference pattern is \(36: 16\). What is the ratio of the widths of the two slits?
A tuning fork $C$ produces $8$ beats per second with another tuning fork D of frequency $340 Hz$. When the prongs of tuning fork $C$ are filed a little, the number of beats produced per second decreases to $4.$ Find the frequency of tuning fork $C$ before filing its prongs.
→For a given medium, the polarizing angle is $60^{\circ}$. What is the critical angle for this medium ?
→