Chapter 12: Q. 13 (page 330)

The four masses shown in FIGURE EX12.13 are connected by massless, rigid rods.
a. Find the coordinates of the center of mass.
b. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page.

Short Answer

Expert verified

The center of mass (x,y) is ( 0.0625, 0.050 )m

Step by step solution

Part (a)

The center of mass x-component

$\begin{array}{l} x_{cm} = \frac{\sum m_{i} x_{i}}{\sum m_{i}} = \frac{m_{A} x_{A} + m_{B} x_{B} + m_{C} x_{C} + m_{D} x_{D}}{m_{A} + m_{B} + m_{C} + m_{D}} \\ x_{cm} = \frac{(100 g) (0 m) + (200 g) (0 m) + (300 g) (0.10 m) + (200 g) (0.10 m)}{100 g + 200 g + 300 g + 200 g} \\ x_{cm} = 0.0625 m \end{array}$

The center of mass y-component

$\begin{array}{l} y_{cm} = \frac{\sum m_{i} y_{i}}{\sum m_{i}} \ \frac{m_{A} y_{A} + m_{B} y_{B} + m_{C} y_{C} + m_{D} y_{D}}{m_{A} + m_{B} + m_{C} + m_{D}} \\ y_{cm} = \frac{(100 g) (0 m) + (200 g) (0.08 m) + (300 g) (0.08 cm) + (200 g) (0 m)}{100 g + 200 g + 300 g + 200 g} \\ y_{cm} = 0.05 m \end{array}$

Part (b)

The distance from the axis to mass C is .

$\sqrt{{(0.10 m)}^{2} + {(0.08 m)}^{2}}$

The moment of inertia through and perpendicular to the page is

$\begin{array}{l} I_{A} = \sum_{i} m_{i} r_{i}^{2} = m_{A} r_{A}^{2} + m_{B} r_{B}^{2} + m_{C} r_{C}^{2} + m_{D} r_{D}^{2} \\ I_{A} = (0.100 kg) {(0 m)}^{2} + (0.200 kg) {(0.08 m)}^{2} + (0.200 kg) {(\sqrt{{(0.10 m)}^{2} + {(0.08 m)}^{2}})}^{2} + (0.200 kg) {(0.10 m)}^{2} \\ I_{A} = 0.0080 kg m \end{array}$

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The four masses shown in FIGURE EX12.13 are connected by massless, rigid rods.
a. Find the coordinates of the center of mass.
b. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page.

Short Answer

Step by step solution

Part (a)

Part (b)

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The four masses shown in FIGURE EX12.13 are connected by massless, rigid rods.a. Find the coordinates of the center of mass.b. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page.

Short Answer

Step by step solution

Part (a)

Part (b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Particle Model of Matter

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The four masses shown in FIGURE EX12.13 are connected by massless, rigid rods.
a. Find the coordinates of the center of mass.
b. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page.