Rotational Inertia, Torque and Angular Acceleration (1Q)

In summary, the angular acceleration of a 5 kg solid sphere with a radius of 0.6 meters when a force of 3 N is applied perpendicular to its radius is 10 rad/sec^2. However, if the sphere were hollow, the moment of inertia would be different and equal to (2/3)mr^2. It is important to note that the moment of inertia for a solid sphere is not equal to mr^2, but instead is equal to (2/5)MR^2.
  • #1
Sam Cepeda
3
0

Homework Statement



. What is the angular acceleration if a force of 3 N is applied perpendicular to the radius of a 5 kg solid sphere that has a radius of 0.6 meters?

Homework Equations


tauv.gif
= r x F
I = mr^2


The Attempt at a Solution


I used
9be08b9254aaacbc0386b26bf137f2ae.png

tried solving for angular acce.
3N/(5kg)(0.6) =10rad/sec^2

Also, if the question had said "hollow sphere" would the moment of inertia be different..(2/3)mr^2 ?
 
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  • #3
Sam Cepeda said:

Homework Statement



. What is the angular acceleration if a force of 3 N is applied perpendicular to the radius of a 5 kg solid sphere that has a radius of 0.6 meters?

Homework Equations


tauv.gif
= r x F
I = mr^2


The Attempt at a Solution


I used
9be08b9254aaacbc0386b26bf137f2ae.png

tried solving for angular acce.
3N/(5kg)(0.6) =10rad/sec^2

Also, if the question had said "hollow sphere" would the moment of inertia be different..(2/3)mr^2 ?

For a solid sphere, I ≠ mR2; I = (2/5) M ⋅ R2

Here is a list of the mass moments of inertia for a variety of different shapes:

https://en.wikipedia.org/wiki/List_of_moments_of_inertia
 
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1. What is rotational inertia and how is it different from mass?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It is similar to mass in that it is a measure of an object's inertia, or tendency to resist changes in motion. However, rotational inertia specifically applies to rotational motion, while mass applies to linear motion.

2. How is torque related to rotational inertia?

Torque is a measure of the force that causes an object to rotate. It is directly proportional to both the force applied and the lever arm, or distance from the pivot point. Rotational inertia plays a role in torque because it determines how difficult it is to change an object's rotational motion, and therefore affects the amount of torque needed to produce a certain amount of rotational acceleration.

3. What is angular acceleration and how is it related to torque?

Angular acceleration is the rate of change of an object's angular velocity, or how quickly it is accelerating or decelerating in its rotational motion. It is directly proportional to the torque applied and inversely proportional to the object's rotational inertia. This means that a higher torque will result in a greater angular acceleration, while a higher rotational inertia will result in a lower angular acceleration.

4. How does the distribution of mass affect an object's rotational inertia?

The distribution of mass in an object affects its rotational inertia because it determines how the mass is distributed around the object's axis of rotation. Objects with more mass concentrated near the axis of rotation will have a lower rotational inertia, while objects with more mass concentrated farther away from the axis of rotation will have a higher rotational inertia. This is why objects like pencils or brooms are easier to rotate than objects like hula hoops or bicycles.

5. What are some real-life applications of rotational inertia, torque, and angular acceleration?

Rotational inertia, torque, and angular acceleration have many practical applications in everyday life. In sports, understanding these concepts is important for maximizing the power and accuracy of movements, such as swinging a baseball bat or throwing a football. In engineering, these concepts are crucial for designing machines and structures that can withstand rotational forces, such as gears and bridges. Additionally, understanding rotational inertia is important in the design of vehicles, as it affects their stability and handling. Finally, these concepts are also important in the study of celestial mechanics, as they help explain the movements of planets and other celestial bodies.

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