Constrained, uneven dumbbell

Classical Mechanics Level 5

Two beads of mass $\displaystyle 2m$ and $\displaystyle m$ , connected by a rod of length $\displaystyle\ell$ and of negligible mass, are free to move in a smooth vertical circular wire frame of radius $\displaystyle\ell$ as shown.

Calculate the velocity (in $\displaystyle m/s$ ) that should be given to the mass $\displaystyle 2m$ (when the rod is in horizontal position) in counter-clockwise direction so that the rod just becomes vertical.

Details and Assumptions:
$\bullet$ $\displaystyle m = 250g$
$\bullet$ $\displaystyle \ell = 25cm$
$\bullet$ $\displaystyle g = 9.8 m/s^2$

Note: This problem appeared in our AITS (All India Test Series) - 6

The answer is 1.851.

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Jatin Yadav
May 8, 2014

Both masses should have same speed so that their separation remains same. So, though not specified, I assume that mass $m$ is also given speed $v$

Use conservation of energy, to form the equation:

$\dfrac{1}{2} 3m v^2 = 3mg(\dfrac{\sqrt{3}}{2}l - \dfrac{l}{6})$ to get the answer as $1.8511$ .

Note If only mass $2m$ is given speed $v$ , then, there would be collision causing loss in energy. We can use conservation of angular momentum about center of loop(as both the impulses on system pass through it) to obtain the new common speed as $v' = \dfrac{2v}{3}$ . Then the answer would have been $v = \dfrac{3}{2} \times 1.8511 = 2.77$

The rod is rigid, so if $2m$ has a velocity $v$ , then $m$ would also have the same velocity.

Anish Puthuraya - 7 years, 1 month ago

I agree with Jatin. It is given that only the mass $2m$ is given a velocity. So immediately after that there would be an impulse due to which both acquire the same speed and there would be a loss of energy in the process.

Karthik Kannan - 7 years, 1 month ago

can we calculate angular accelaration of the rod and blocks system about center of circle. by net torque applied by gravity on both blocks...and then using equation of rotational motion only for "2m", for which we already know by how much it(2m) has rotated ,we also know the angular acclaration of the system and specifically we also know that final angular velocity which is 0. So we can calculate the initial angular velocity given and from that we can calculate the intial velocity......please tell if iam correct because my answer is 1.96.

Naman Negi - 7 years ago

how did we arrive at the given equation using energy conservation?

Athul Nambolan - 7 years, 1 month ago

use the center of mass motion

Beakal Tiliksew - 7 years, 1 month ago

so answer should be 2.77 m/s

Shubham Maurya - 7 years ago

hw cn yu say dat both masses vll have same speed.......velocity of 2m and m along de rod and perpendiculur to de rod shud be same......afte dat apply energy conservation....ull get properly 1.84 metres per second

sid leo - 7 years ago

Constrained, uneven dumbbell

The answer is 1.851.

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