Chapter 12: Q. 14 (page 330)

14. II 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 a diagonal axis that passes through masses B and D.

Short Answer

Expert verified

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

b. The moment of inertia about a diagonal is $0.00163 kg m^{2}$

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 length of the diagonal is,

$d = \sqrt{{(10 cm)}^{2} + {(8 cm)}^{2}} = 12.8 cm$

The distance at point and point is

$\begin{array}{l} r = \frac{d}{2} \\ r = \frac{12.8 cm}{2} = 6.4 cm \\ r_{A} = r_{C} = 6.4 cm \end{array}$

As the axis is passing through the and , so, the distance at point and is zero.

$r_{B} = r_{D} = 0$

The moment of inertia about a diagonal that passes through B and D is

$\begin{array}{l} I_{BD} = m_{A} r_{A}^{2} + m_{C} r_{C}^{2} + m_{B} r_{B}^{2} + m_{D} r_{D}^{2} \\ I_{BD} = (0.100 kg) {(0.064 m)}^{2} + (0.300 kg) {(0.064 m)}^{2} + 0 + 0 \\ I_{BD} = 0.00163 kg m^{2} \end{array}$

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14. II 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 a diagonal axis that passes through masses B and D.

Short Answer

Step by step solution

Part (a)

Part (b)

One App. One Place for Learning.

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14. II 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 a diagonal axis that passes through masses B and D.

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

Turning Points in Physics

Physics of Motion

Translational Dynamics

Conservation of Energy and Momentum

Dynamics

Space Physics

Study anywhere. Anytime. Across all devices.

14. II 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 a diagonal axis that passes through masses B and D.