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Chapter 10: Problem 97

If the body's center of mass were not placed on the rotational axis of the turntable, how would the person's measured moment of inertia compare to the moment of inertia for rotation about the center of mass? (a) The measured moment of inertia would be too large; (b) the measured moment of inertia would be too small; (c) the two moments of inertia would be the same; (d) it depends on where the body's center of mass is placed relative to the center of the turntable.

Short Answer

Expert verified
The measured moment of inertia would be too large (option a).

Step by step solution

01

Understand the Concept of Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. It depends on the mass distribution relative to the axis of rotation. If mass is farther from the axis, the moment of inertia increases.
02

Apply the Parallel Axis Theorem

The parallel axis theorem states that the moment of inertia about any axis parallel to one through the center of mass is given by: \( I = I_{cm} + Md^2 \), where \( I_{cm} \) is the moment of inertia through the center of mass, \( M \) is the mass, and \( d \) is the distance between the axes.
03

Analyze the Problem

In this problem, the body's center of mass is away from the rotational axis of the turntable. According to the parallel axis theorem, the moment of inertia measured around the turntable's axis will be \( I = I_{cm} + Md^2 \), which is greater than \( I_{cm} \) since \( Md^2 > 0 \).
04

Conclusion

The measured moment of inertia is greater when the axis of rotation is not through the center of mass. Therefore, the correct answer is (a) The measured moment of inertia would be too large.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Center of Mass
The center of mass (COM) of an object is a critical point where the entire mass seems to be concentrated. It is the average location of all the mass in a body. Imagine it as the balancing point of an object where it would stay level if supported.
This concept is crucial in understanding rotational movements and balance.
  • If a force is applied at the center of mass, the object will move without rotating.
  • If the force is applied elsewhere, it will cause the object to rotate around the center of mass.
Understanding the center of mass is essential when discussing the moment of inertia, as it affects how easily an object can be spun around an axis.
Parallel Axis Theorem
The parallel axis theorem is a fundamental concept in rotational dynamics. It provides a way to calculate an object's moment of inertia when the rotational axis is not through the center of mass. According to this theorem:
  • The moment of inertia about an axis parallel to one through the center of mass can be calculated with the equation: \[ I = I_{cm} + Md^2 \]
  • \( I_{cm} \) is the moment of inertia through the center of mass.
  • \( M \) is the total mass of the object.
  • \( d \) is the distance between the two axes.
This theorem shows that the moment of inertia increases when the axis is farther from the center of mass. Hence, the farther the rotational axis is from the center of mass, the larger the contribution of \( Md^2 \) to the total moment of inertia.
Rotational Axis
The rotational axis is the line around which an object rotates. The position of this axis plays a significant role in the rotational dynamics of the body.
  • It can pass through the center of mass, leading to minimal resistance to rotation.
  • Or it can be offset, causing increased resistance due to the need for more force to rotate the object.
Choosing the right axis of rotation is crucial in engineering and design, as it affects the effort needed to spin or rotate objects. If the axis does not pass through the center of mass, the object will experience a larger moment of inertia, making it more resistant to changes in its rotational motion.
Mass Distribution
Mass distribution refers to how mass is spread throughout an object. It has a direct impact on the moment of inertia. The further the mass is from the rotational axis, the greater the moment of inertia will be.
  • For a concentrated mass closer to the rotational axis, the moment of inertia is lower.
  • For a spread-out mass farther from the axis, the moment of inertia increases significantly.
This idea is essential in understanding why the moment of inertia can change with different mass arrangements. Engineers and designers must consider mass distribution in their work, especially when designing rotating systems like wheels or turbines.

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Most popular questions from this chapter

The moment of inertia of the empty turntable is \(1.5 \mathrm{~kg} \mathrm{~m}^{2}\). With a constant torque of \(2.5 \mathrm{~N} \cdot \mathrm{m},\) the turntable-person system takes \(3.0 \mathrm{~s}\) to spin from rest to an angular speed of \(1.0 \mathrm{rad} / \mathrm{s}\). What is the person's moment of inertia about an axis through her center of mass? Ignore friction in the turntable axle. (a) \(2.5 \mathrm{~kg} \cdot \mathrm{m}^{2}\) (b) \(6.0 \mathrm{~kg} \cdot \mathrm{m}^{2}\) (c) \(7.5 \mathrm{~kg} \cdot \mathrm{m}^{2}\) (d) \(9.0 \mathrm{~kg} \cdot \mathrm{m}^{2}\).

A hollow, thin-walled sphere of mass 12.0 kg and diameter 48.0 cm is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by \(\theta(t) = At^2 + Bt^4\), where \(A\) has numerical value 1.50 and \(B\) has numerical value 1.10. (a) What are the units of the constants \(A\) and \(B\)? (b) At the time 3.00 s, find (i) the angular momentum of the sphere and (ii) the net torque on the sphere.

A thin uniform rod has a length of 0.500 m and is rotating in a circle on a frictionless table. The axis of rotation is perpendicular to the length of the rod at one end and is stationary. The rod has an angular velocity of 0.400 rad/s and a moment of inertia about the axis of 3.00 \(\times 10^{-3}\) kg \(\cdot\) m\(^2\). A bug initially standing on the rod at the axis of rotation decides to crawl out to the other end of the rod. When the bug has reached the end of the rod and sits there, its tangential speed is 0.160 m/s. The bug can be treated as a point mass. What is the mass of (a) the rod; (b) the bug?

A cord is wrapped around the rim of a solid uniform wheel 0.250 m in radius and of mass 9.20 kg. A steady horizontal pull of 40.0 N to the right is exerted on the cord, pulling it off tangentially from the wheel. The wheel is mounted on frictionless bearings on a horizontal axle through its center. (a) Compute the angular acceleration of the wheel and the acceleration of the part of the cord that has already been pulled off the wheel. (b) Find the magnitude and direction of the force that the axle exerts on the wheel. (c) Which of the answers in parts (a) and (b) would change if the pull were upward instead of horizontal?

A uniform marble rolls down a symmetrical bowl, starting from rest at the top of the left side. The top of each side is a distance \(h\) above the bottom of the bowl. The left half of the bowl is rough enough to cause the marble to roll without slipping, but the right half has no friction because it is coated with oil. (a) How far up the smooth side will the marble go, measured vertically from the bottom? (b) How high would the marble go if both sides were as rough as the left side? (c) How do you account for the fact that the marble goes \(higher\) with friction on the right side than without friction?
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