Ch 12: Rotation of a Rigid Body

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 12: Rotation of a Rigid Body

Problem 49a

Chapter 12, Problem 49a

A small 300 g ball and a small 600 g ball are connected by a 40-cm-long, 200 g rigid rod. a. How far is the center of mass from the 600 g ball?

Verified step by step guidance

Step 1: Understand the concept of the center of mass. The center of mass is the weighted average position of all the masses in a system. For a system of discrete masses, the formula to calculate the center of mass is:

x = \frac{\sum_{i}^{n} m x_{i}}{\sum_{i}^{n}}

, where

m

is the mass and

x_{i}

is the position of each mass.

Step 2: Assign positions to the masses. Let the position of the 600 g ball be at

x = 0

. The position of the 300 g ball is at

x = 40

cm (the length of the rod). The rod itself has a mass of 200 g, and its center of mass is located at its midpoint, which is

x = 20

cm.

Step 3: Write the formula for the center of mass of the system. The total mass of the system is the sum of all individual masses:

M = 600 + 300 + 200 = 1100

g. The center of mass is given by:

x = \frac{600 \times 0 + 300 \times 40 + 200 \times 20}{1100}

Step 4: Simplify the numerator of the center of mass formula. Multiply each mass by its respective position:

600 \times 0 = 0

300 \times 40 = 12000

, and

200 \times 20 = 4000

. Add these values together to get the total numerator.

Step 5: Divide the total numerator by the total mass to find the center of mass. The result will give the position of the center of mass relative to the 600 g ball. Ensure the units are consistent throughout the calculation (e.g., grams and centimeters).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Center of Mass

The center of mass is a point that represents the average position of the mass distribution of a system. It is calculated by taking into account the masses of the objects and their distances from a reference point. For a system of particles, the center of mass can be found using the formula: x_cm = (Σ(m_i * x_i)) / Σm_i, where m_i is the mass and x_i is the position of each particle.

Mass Distribution

Mass distribution refers to how mass is spread out in a system. In this problem, we have two balls with different masses connected by a rigid rod, which affects the overall center of mass. Understanding how the masses are arranged relative to each other is crucial for calculating the center of mass accurately.

Rigid Body

A rigid body is an idealization in physics where an object does not deform under stress, meaning the distances between its constituent particles remain constant. In this scenario, the rigid rod connecting the two balls ensures that their relative positions do not change, simplifying the calculation of the center of mass as we can treat the system as a single entity.

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Related Practice

Textbook Question

A 75 g, 6.0-cm-diameter solid spherical top is spun at 1200 rpm on an axle that extends 1.0 cm past the edge of the sphere. The tip of the axle is placed on a support. What is the top's precession frequency in rpm?

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Textbook Question

A small 300 g ball and a small 600 g ball are connected by a 40-cm-long, 200 g rigid rod. b. What is the rotational kinetic energy if the structure rotates about its center of mass at 100 rpm?

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Textbook Question

Calculate by direct integration the moment of inertia for a thin rod of mass M and length L about an axis located distance d from one end. Confirm that your answer agrees with Table 12.2 when d=0 and when d = L/2.

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Textbook Question

Determine the moment of inertia about the axis of the object shown in FIGURE P12.52.

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