Math Problem Statement
Solution
To solve this problem, we need to use the conservation of angular momentum. Here's the detailed breakdown:
Given Information:
- Two masses, each , are attached to a rigid rod of negligible mass.
- The rod is pivoted at its center, with the masses located on each side from the pivot.
- A piece of putty is dropped on the right mass at a speed of and sticks to it.
Required:
The angular velocity () of the system just after the putty sticks.
Solution:
-
Calculate the initial angular momentum of the putty relative to the pivot point: where:
- (distance from the pivot to the mass where the putty sticks)
Substituting these values:
-
Calculate the moment of inertia (I) of the system after the putty sticks:
- Each mass is from the pivot.
- The putty, now stuck to the right mass, also contributes to the moment of inertia at from the pivot.
So, Breaking this down:
-
Use Conservation of Angular Momentum: Since there is no external torque, the initial angular momentum of the putty must equal the final angular momentum of the entire system.
Solving for :
Answer:
The angular velocity of the masses just after the putty hits and sticks to the right mass is:
This matches option (B) in the problem.
Would you like further details on any part of the solution? Here are some follow-up questions you may find helpful:
- How does the conservation of angular momentum apply to rotating systems?
- Why is the moment of inertia calculated based on distances from the pivot point?
- How would the result change if the rod itself had a non-negligible mass?
- What would happen if the putty didn’t stick and instead bounced off?
- How do we handle angular momentum when forces are applied at different angles?
Tip: When using conservation laws, ensure you account for all masses and their respective distances from the pivot to correctly calculate the moment of inertia.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Conservation of Angular Momentum
Moment of Inertia
Angular Velocity
Formulas
L = Iω
L = mvr
I = Σmr²
Theorems
Conservation of Angular Momentum
Suitable Grade Level
Grade 12 or College Level
Related Recommendation
Angular Momentum and Moment of Inertia in Rotational Dynamics
Conservation of Angular Momentum: Disk and Rod Angular Velocity Calculation
Collision of Stick and Putty: Angular Velocity and Initial Speed
Angular Momentum of Rotating Discs Connected by a Rigid Rod
Calculate Angular Velocity of a Rotating Rod After Collision with Ball