Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the angular momentum vector of the two particle system is the same whatever be the point about which the angular momentum is taken.
Download our app for free and get started
Let at a certain instant two particles be at points P and Q, as shown in the following figure:
Angular momentum of the system about point $P$,
$L_p=m v \times 0+m v \times d=m v d \ldots$...(i) Angular momentum of the system about point $Q L_Q=m v \times d+m v \times 0=m v d \ldots$
(ii) Consider a point $R$, which is at a distance $y$ from point $Q$ i.e., $Q R=y \therefore P R=d-y$ Angular momentum of the system about point $R, L_R=m v \times(d-y)+m v \times y m v d-m v y+m v y=m v d \ldots$...(iii) Comparing equations (i), (ii), and (iii), we get, $L_p=L_Q=L_R \ldots$...(iv) We infer from equation (iv) that the angular momentum of a system does not depend on the point about which it is taken.
Angular momentum of the system about point $P$,
$L_p=m v \times 0+m v \times d=m v d \ldots$...(i) Angular momentum of the system about point $Q L_Q=m v \times d+m v \times 0=m v d \ldots$
(ii) Consider a point $R$, which is at a distance $y$ from point $Q$ i.e., $Q R=y \therefore P R=d-y$ Angular momentum of the system about point $R, L_R=m v \times(d-y)+m v \times y m v d-m v y+m v y=m v d \ldots$...(iii) Comparing equations (i), (ii), and (iii), we get, $L_p=L_Q=L_R \ldots$...(iv) We infer from equation (iv) that the angular momentum of a system does not depend on the point about which it is taken.
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1View SolutionFind the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components px, py and pz. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.
- 2A grindstone has a moment of inertia of $6kg m^2$. A constant torque is applied and the grindstone is found to have a speed of $150rpm, 10s$. after starting from rest. Calculate the torque.View Solution
- 3View SolutionA particle of mass m is projected from origin O with speed u at an angle with positive x-axis. Find the angular momentum of particle at any time t about O before the particle strikes the ground again.
- 4Two blocks of masses $400g$ and $200g$ are connected through a light string going over a pulley which is free to rotate about its axis. The pulley has a moment of inertia $1.6 \times 10^{-4}kg-m^2$ and a radius 2.0cm. Find:View Solution
- The kinetic energy of the system as the $400g$ block falls through $50cm$.
- The speed of the blocks at this instant.
- 5View SolutionThe structure of a water molecule is shown in figure. Find the distance of the centre of mass of the molecule from the centre of the oxygen atom.
- 6A hollow sphere is released from the top of an inclined plane of inclination $\theta.$View Solution
- What should be the minimum coefficient of friction between the sphere and the plane to prevent sliding?
- Find the kinetic energy of the ball as it moves down a length 1 on the incline if the friction coefficient is half the value calculated in part (a).
- 7A non-uniform bar of weight W is suspended at rest by two strings of negligible weight as shown in. The angles made by the strings with the vertical are $36.9°$ and $53.1°$ respectively. The bar is $2m$ long. Calculate the distance d of the centre of gravity of the bar from its left end.View Solution
- 8View SolutionA wheel in uniform motion about an axis passing through its centre and perpendicular to its plane is considered to be in mechanical (translational plus rotational) equilibrium because no net external force or torque is required to sustain its motion. However, the particles that constitute the wheel do experience a centripetal acceleration directed towards the centre.How do you reconcile this fact with the wheel being in equilibrium? How would you set a half-wheel into uniform motion about an axis passing through the centre of mass of the wheel and perpendicular to its plane? Will you require external forces to sustain the motion?
- 9View SolutionFind the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components px, py and pz. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.
- 10In the HCl molecule, the separation between the nuclei of the two atoms is about $1.27\mathring{\text{A}}\big(1\mathring{\text{A}}= 10^{-10} \text{m}\big).$ Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.View Solution