Qn. on general rotational mechanics

[manogna08]

Homework Statement

a uniform rod of mass m nd length 2l lies on smooth horizontal surface. a particle of mass m is connected to a string of length l whose other end is connected to one of the ends of the rod. initially string is taut and both string and rod lie in same plane with 90 angle b/w them. if particle is given velocity v perpendicular to string, then jus after givin velocity find tension in string and angular acceleration of rod.

Homework Equations

The Attempt at a Solution

i tried using ζ = Iα about centre of mass but could not succeed..pleasez help...

 

[lewando]

I would think that when the velocity was imparted to the particle, since is it attached to the rod via a string (assumed inextensible, otherwise it's a spring) then the same velocity would be imparted to the end of the rod.

 

[haruspex]

I would think that when the velocity was imparted to the particle, since is it attached to the rod via a string (assumed inextensible, otherwise it's a spring) then the same velocity would be imparted to the end of the rod.

No, the particle is driven perpendicularly to the string.
Always start by creating some names for unknowns that might be relevant:
T = tension
α = angular acceleration of rod
a = linear acceleration of rod.
Then draw the free body diagram and try to write down some equations using the conservation laws. Post whatever you come up with.

 

[lewando]

That makes for a more interesting problem. Reading is fundamental .

 

[Nickie]

Dear Student,

Thank you for your question. It seems like you are on the right track by using the equation ζ = Iα about the center of mass. However, in order to solve this problem, you will need to use a combination of rotational and translational equations.

First, let's define some variables:
m = mass of the rod and particle
l = length of the string
v = velocity of the particle
t = tension in the string
α = angular acceleration of the rod

Now, let's start by looking at the translational motion of the particle. Since the particle is moving in a circle with radius l, we can use the equation for centripetal force:
F = mv^2/l

This force is provided by the tension in the string, so we can set it equal to t:
t = mv^2/l

Next, we can look at the rotational motion of the rod. The rod will experience a torque due to the tension in the string, which will cause it to rotate. The equation for torque is τ = Iα, where I is the moment of inertia of the rod. Since the rod is rotating about its center of mass, we can use the equation for a rod rotating about its center:
I = (1/12)ml^2

Substituting this into the torque equation, we get:
t*l = (1/12)ml^2 * α

Now, we have two equations for t, so we can set them equal to each other and solve for α:
mv^2/l = (1/12)ml^2 * α
α = 12v^2/l^3

Finally, we can substitute this value of α into either of the equations for t to find the tension in the string:
t = (1/12)ml^2 * α/l
t = 12mv^2/l^2

Therefore, the tension in the string is 12 times the mass of the particle times its velocity squared divided by the length of the string squared.

I hope this helps you solve the problem. Remember to always think about both translational and rotational motion when dealing with problems involving rotating objects. Good luck!

Sincerely,