Chapter 8: Problem 23
How does increasing the moment of inertia affect how easily an object rotates?
Short Answer
Expert verified
Increasing the moment of inertia makes it harder for an object to rotate.
Step by step solution
01
Understand the Moment of Inertia
The moment of inertia is a measure of an object's resistance to change in its rotational motion. It depends on the mass distribution relative to the axis of rotation. For an object to rotate, a torque must overcome this resistance.
02
Establish the Relationship
The rotational analog of Newton's Second Law is given by \( \tau = I \alpha \), where \( \tau \) is the torque, \( I \) is the moment of inertia, and \( \alpha \) is the angular acceleration. This equation shows that for a given torque, the angular acceleration is inversely proportional to the moment of inertia.
03
Analyze the Effect of Increasing Moment of Inertia
As the moment of inertia \( I \) increases, for the same amount of applied torque \( \tau \), the angular acceleration \( \alpha \) decreases. This means that it becomes harder for the object to start or change its rotational motion.
04
Compare with Decreasing Moment of Inertia
Conversely, if the moment of inertia decreases, the same torque will produce a larger angular acceleration, making it easier for the object to rotate or change its rotational state.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rotational Motion
Think of rotational motion as a spin or turn around an axis. Instead of moving in a straight line, objects in rotational motion move along a circular path. A perfect example is a spinning top. Each part of the top travels in a circle either small or large, depending on its distance from the center.
The center point or line around which everything spins is called the axis of rotation. In our spinning top example, it's the very thin line running through its middle.
The center point or line around which everything spins is called the axis of rotation. In our spinning top example, it's the very thin line running through its middle.
- Rotational motion involves angular quantities like angular velocity.
- Unlike linear motion, distance in rotational movement is measured in angles (radians).
- The speed of rotation is termed angular speed, and it tells us how fast an object is spinning.
Torque
Torque is essentially the rotational counterpart of force. Just as force makes objects move in a straight line, torque makes them spin or rotate. Think of it as a twist you give to something.
Imagine using a wrench to tighten a bolt. When you apply force at the handle, you're creating torque. The farther out from the bolt head you apply the force, the greater the torque.
Imagine using a wrench to tighten a bolt. When you apply force at the handle, you're creating torque. The farther out from the bolt head you apply the force, the greater the torque.
- Torque is calculated using the formula: \( \tau = rF\sin\theta \), where \( r \) is the lever arm length, \( F \) is the applied force, and \( \theta \) is the angle between \( F \) and \( r \).
- This means the effectiveness of the applied force depends not just on its magnitude, but also on its direction and point of application.
Angular Acceleration
Angular acceleration is how the speed of rotation changes over time. It's similar to how a speeding car gains speed, but in a rotational context. If a spinning wheel turns faster, it's experiencing angular acceleration. When torque is applied to an object, it doesn't just start spinning; it accelerates. This change in spinning speed is quantified as angular acceleration.
- The relationship is captured by \( \alpha = \frac{\tau}{I} \), where \( \alpha \) is angular acceleration, \( \tau \) is the torque, and \( I \) is the moment of inertia.
- This tells us angular acceleration relies on both the torque and the resistance to rotation (moment of inertia).