Chapter 11: Problem 1
A 0.120-kg, 50.0-cm-long uniform bar has a small 0.055-kg mass glued to its left end and a small 0.110-kg mass glued to the other end. The two small masses can each be treated as point masses. You want to balance this system horizontally on a fulcrum placed just under its center of gravity. How far from the left end should the fulcrum be placed?
Short Answer
Step by step solution
Determine the Position of the Center of the Uniform Bar
Calculate the Moments about a Chosen Point
Calculate the Total Moment about Point O
Calculate the Total Mass of the System
Calculate the Center of Gravity from the Left End
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Uniform Bar
The uniform distribution of mass allows us to pinpoint the center of gravity (CG) easily. We calculate it to be precisely at the geometric center of the bar. For example, if a bar is 50.0 cm long, its CG will be at 25.0 cm from either end.
This concept helps us to understand how the weight of the bar is balanced. Such an understanding is crucial when performing further calculations like finding moments and balancing the bar on a fulcrum.
Moment of Inertia
For a point mass, the moment of inertia can be calculated simply by multiplying the mass by the square of the distance from the pivot:
- For a point mass, the formula is given as: \[ I = m imes r^2 \]where \( I \) is the moment of inertia, \( m \) is the mass, and \( r \) is the distance from the distance to the axis of rotation.
By understanding the moment of inertia, we can calculate the total effects of all masses on the system's rotation.
Point Masses
In a physics problem, each point mass affects the balance of the system. We calculate the contribution of each point mass using their masses and their distances from a reference point. For example:
- A 0.055-kg point mass on the left contributes minimally since it rests very close to the reference point.
- A 0.110-kg mass on the right contributes significantly more as it affects the balance further away from the pivot.
Fulcrum Balance
The balance point, or the fulcrum, is typically placed directly beneath the center of gravity. In achieving balance, the sum of clockwise moments must equal the sum of counterclockwise moments around the fulcrum.
- Moments are calculated by multiplying the weight of each mass with its distance from the fulcrum.
- Adjusting the fulcrum position can re-distribute these moments to maintain equilibrium.