Q: Given a thin, flat plate with a continuous mass distribution, which formula gives the $$y-$$coordinate of the center of mass?

$$\displaystyle \bar{y} = \frac{\int \tilde{y} dm}{\int dm}$$

Question 1

A uniform thin plate in the shape of a region bounded by $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$ has density $$\delta. $$Which formula gives the $$x-$$coordinate of its center of mass?

Accepted Answer

$$\displaystyle \bar{x} = \frac{1}{M} \int_a^b \delta x [f(x) - g(x)] dx$$

Question 2

Given two masses $$m_1$$ and $$m_2$$ located at $$x_1$$ and $$x_2 on a $$rigid, weightless rod, what is the formula for the center of mass $$\bar{x}$$?

Accepted Answer

$$\displaystyle \bar{x} = \frac{x_1$ m_1$ + x_2$ m_2$}{m_1$ + m_2$}$$

Question 3

A seesaw is balanced with a child of weight $$W_1$$ sitting $$d_1$$ meters from the fulcrum and another child of weight $$W_2$$ sitting $$d_2$$ meters on the other side. Which equation must be satisfied for balance?

Accepted Answer

$$d_1$$ $$W_1$$ = $$d_2$$ $$W_2$$

Question 4

For a continuous mass distribution with density $$\delta(x, y)$$ over a region $$R$$, the total mass $$M is $$given by:

Accepted Answer

$$\displaystyle M = \int_R \delta(x, y) dA$$

Question 5

If a plate has a constant density $$\delta$$ and occupies the region between $$y = 0$$ and $$y = 2x$$ for $$0 \leq x \leq 1$$, what is the area of the plate?

Accepted Answer

$$\displaystyle \$$int_0$$^1 2x dx = 1$$

Question 6

Which of the following best describes a rigid body in the context of center of mass calculations?

Accepted Answer

A body whose shape cannot be changed.

Question 7

Given a thin, flat plate with a continuous mass distribution, which formula gives the $$y-$$coordinate of the center of mass?

Accepted Answer

$$\displaystyle \bar{y} = \frac{\int 	ilde{y} dm}{\int dm}$$

Question 8

If the density $$\delta(x, y) is $$constant, how does the $$x-$$coordinate of the center of mass $$\bar{x}$$ depend on the plate?

Accepted Answer

$$\bar{x}$$ depends only on the shape of the plate.

Question 9

For a region bounded by $$y = f(x)$$ and $$y = g(x)$$, $$a \leq x \leq b$$, with constant density $$\delta$$, what is the formula for the total mass $$M$$?

Accepted Answer

$$\displaystyle M = \int_a^b \delta [f(x) - g(x)] dx$$

Question 10

A triangular plate with vertices at $$(0,0)$$, $$(1,0)$$, and $$(1,2)$$ has constant density $$\delta = 3 g/cm$^2. $$What is the $$x-$$coordinate of its center of mass?

Accepted Answer

$$\displaystyle \frac{2}{3} cm$$

Question 11

If the density $$\delta(x, y) is a $$continuous function of $$x$$, then the center $$(	ilde{x}, 	ilde{y}) of a $$thin vertical strip is located:

Accepted Answer

At the midpoint of the strip.

Question 12

The density at a point $$(x, y) in a $$plate is defined as:

Accepted Answer

$$\displaystyle \delta(x, y) = \lim_{\Delta A 	o 0} \frac{\Delta m}{\Delta A}$$

Question 13

Which of the following is true about the connections between masses in a rigid system for center of mass calculations?

Accepted Answer

The connections have negligible mass compared to the masses in the system.

Question 14

Suppose a plate occupies the region between $$y = 0$$ and $$y = 2x$$ for $$0 \leq x \leq 1$$ and has density $$\delta = 3 g/cm$^2. $$What is the total mass $$M of $$the plate?

Accepted Answer

$$3 g$$

Question 15

For a region bounded by $$y = f(x)$$ and $$y = g(x)$$, $$a \leq x \leq b$$, with constant density $$\delta$$, what is the formula for the $$y-$$coordinate of the center of mass?

Accepted Answer

$$\displaystyle \bar{y} = \frac{1}{M} \int_a^b \delta \frac{1}{2} [f^2$(x) - g^2$(x)] dx$$

Question 16

If an object is cut into several pieces, each with its own center of mass, how can the overall center of mass be found?

Accepted Answer

By treating each piece as a point mass at its center of mass and using the center of mass formula.

Question 17

Which of the following statements is true regarding the calculation of the center of mass for a plate with variable density?

Accepted Answer

The center of mass depends on both the shape and the density distribution.

6.6 Moments and Centers of Mass

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