If the ends of a rod of length l have velocities v → A and v → B , find the velocity of mid-point of the rod.

Let us assume that, the mid-point of the rod has velocity v → C . Then, we have v → A = v → A C + v → C . . . i v → B = v → B C + v → C . . . i i Adding Eqs. (i) and (ii), we have v → A + v → B = v → A C + v → B C + 2 v → C Substituting v → A C = + ω → A C × r → A C and v → B C = + ω → B C × r → B C , we get v → A + v → B = ω → A C × r → A C + ω → B C × r → B C + 2 v → C . Substituting ω → A C = ω → B C = ω → (angular velocity of the rod), we have, v → A + v → B = ω → × r → A C + r → B C + 2 v → C . Since C is the mid-point of the rod, r → A C = - r → B C . Then we have, v → C = v → A + v → B 2 This tells us that when any two points A and B , (say;) of a rigid body have velocities v → A and v → B respectively, the mid-point C (but not necessarily the centre of mass) of the line joining the points will have a velocity v → C = v → A + v → B 2 .

If the ends of a rod of length l have velocities v → A and v → B , find the velocity of mid-point of the rod.

Let us assume that, the mid-point of the rod has velocity v → C . Then, we have v → A = v → A C + v → C . . . i v → B = v → B C + v → C . . . i i Adding Eqs. (i) and (ii), we have v → A + v → B = v → A C + v → B C + 2 v → C Substituting v → A C = + ω → A C × r → A C and v → B C = + ω → B C × r → B C , we get v → A + v → B = ω → A C × r → A C + ω → B C × r → B C + 2 v → C . Substituting ω → A C = ω → B C = ω → (angular velocity of the rod), we have, v → A + v → B = ω → × r → A C + r → B C + 2 v → C . Since C is the mid-point of the rod, r → A C = - r → B C . Then we have, v → C = v → A + v → B 2 This tells us that when any two points A and B , (say;) of a rigid body have velocities v → A and v → B respectively, the mid-point C (but not necessarily the centre of mass) of the line joining the points will have a velocity v → C = v → A + v → B 2 .

The rod of length l = 1 m rotates with an angular velocity ω = 2 rad s - 1 and the point P moves with velocity v = 1 m s - 1 and acceleration a = 1 m s - 2 . Find the velocity and acceleration of Q .

Velocity of point Q is given as v Q = v → Q P + v → P = l 2 ω 2 + v 2 + 2 v l ω cos 120 ° ⇒ v Q = ( 1 × 2 ) 2 + 1 2 + 2 ( 1 ) ( 2 ) × - 1 2 = 3 m s - 1 Acceleration of point Q is given as a Q = a → Q P + a → P = a Q P 2 + a P 2 + 2 a Q P a P cos 30 ° where a Q P = l ω 2 = 1 ( 2 ) 2 = 4 ms - 2 and a P = 1 m s - 2 ⇒ a Q = 4 2 + 1 2 + 2 . 4 × 1 × 3 2 = 17 + 4 3 ms - 2

Two wheels 1 and 2 of radii r 1 and r 2 are connected by a belt. If the belt does not slide over the wheels, find the: (i) speed of a point on the belt. (ii) angular velocity of the wheel 2 . (iii) acceleration of P relative to Q .

Given, Radius of two wheels are r 1 and r 2 Speed of a point on the belt (a) As belt is not sliding v A = ω r 1 = v B Angular velocity of the wheel 2 (ii) v B = ω 2 r 2 ⇒ ω 2 - ω r 1 r 2 Acceleration of P relative to Q (iii) a → P = r 1 ω 2 i ^ and a → Q = r 2 ω 2 2 - i ^ ⇒ a → Q = r 2 ω r 1 r 2 2 - i ^ = - ω 2 r 1 2 r 2 i ^

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If the ends of a rod of length $l$ have velocities ${\vec{v}}_{A}$ and ${\vec{v}}_{B},$ find the velocity of mid-point of the rod.

Important Questions on Rigid Body Dynamics: Part 1

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Physics For Joint Entrance Examination JEE (Advanced): Mechanics II>Chapter 3 - Rigid Body Dynamics: Part 1>CONCEPT APPLICATION EXERCISE>Q 5

A flat rigid body is moving in $x - y$ plane on a table. The body lies in the $x - y$ plane. At an instant it was found that some of the velocity components of its three particles $A,$ $B$ and $C$ were $v_{A x} = 4 m / s, v_{B x} = 3 m / s$ and $v_{c y} = - 2 m / s$ respectively. At the instant the three particles $A, B$ and $C$ were located at $(0, 0), (3, 4)$ and $(4, 3)$ (all in metre) respectively in a co-ordinate system attached to the table.

(a) Find the velocity of $A, B$ and $C .$
(b) Find the angular velocity of the body.

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Physics For Joint Entrance Examination JEE (Advanced): Mechanics II>Chapter 3 - Rigid Body Dynamics: Part 1>CONCEPT APPLICATION EXERCISE>Q 6

The rod of length $l = 1 m$ rotates with an angular velocity $A$ and the point $P$ moves with velocity $v = 1 m s^{- 1}$ and acceleration $a = 1 m s^{- 2} .$ Find the velocity and acceleration of $Q .$

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Physics For Joint Entrance Examination JEE (Advanced): Mechanics II>Chapter 3 - Rigid Body Dynamics: Part 1>CONCEPT APPLICATION EXERCISE>Q 7

The angular velocity and angular acceleration of the pivoted rod are given as $ω$ and $α$ respectively. Find the $x$ and $y$ components of acceleration of $B .$

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Physics For Joint Entrance Examination JEE (Advanced): Mechanics II>Chapter 3 - Rigid Body Dynamics: Part 1>CONCEPT APPLICATION EXERCISE>Q 1

A cylinder of radius $5 m$ rolls on a horizontal surface. If the velocity of its center is $25 m / s .$ Find its angular velocity and velocity of the point $A .$

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Physics For Joint Entrance Examination JEE (Advanced): Mechanics II>Chapter 3 - Rigid Body Dynamics: Part 1>CONCEPT APPLICATION EXERCISE>Q 2

A disc of radius $R = \frac{1}{2} m$ is rolling on a plank. If the disc is moving on the plank with angular velocity $ω = 2 rad / s$ and speed $v_{c} = 2 m / s .$ Find the speed of the plank for pure rolling.

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Physics For Joint Entrance Examination JEE (Advanced): Mechanics II>Chapter 3 - Rigid Body Dynamics: Part 1>CONCEPT APPLICATION EXERCISE>Q 3

A uniform disc of radius $R$ rolls perfectly over two horizontal plank $A$ and $B$ moving with velocities $v$ and $2 v,$ respectively. Find the
(a) velocity of $CM$ of the disc.
(b) angular velocity of the disc

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Physics For Joint Entrance Examination JEE (Advanced): Mechanics II>Chapter 3 - Rigid Body Dynamics: Part 1>CONCEPT APPLICATION EXERCISE>Q 4

A cotton reel rolls without sliding such that the point $P$ of the string has velocity $v .$ Find the:
(i) velocity of its centre $C$
(ii) angular velocity of the cotton reel.

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Physics For Joint Entrance Examination JEE (Advanced): Mechanics II>Chapter 3 - Rigid Body Dynamics: Part 1>CONCEPT APPLICATION EXERCISE>Q 5

Two wheels $1$ and $2$ of radii $r_{1}$ and $r_{2}$ are connected by a belt. If the belt does not slide over the wheels, find the:

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(i) speed of a point on the belt.
(ii) angular velocity of the wheel $2 .$
(iii) acceleration of $P$ relative to $Q .$

If the ends of a rod of length l have velocities v→A and v→B, find the velocity of mid-point of the rod.

Important Questions on Rigid Body Dynamics: Part 1

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If the ends of a rod of length $l$ have velocities ${\vec{v}}_{A}$ and ${\vec{v}}_{B},$ find the velocity of mid-point of the rod.