Problem 8 - Calculus Early Transcendentals

Question

Sketch the following systems on a number line and find the location of the center of mass. $$\begin{aligned} &m_{1}=8 \mathrm{kg} \text { located at } x=2 \mathrm{m} ; m_{2}=4 \mathrm{kg} \text { located at } x=-4 \mathrm{m}\\\ &m_{3}=1 \mathrm{kg} \text { located at } x=0 \mathrm{m} \end{aligned}$$

Step-by-Step Solution

Answer

Answer: The center of mass for this system is at x=0m on the number line.

Step 1: Sketch the system

First, let's plot each mass (m1, m2, and m3) on a number line. Place m1 at x=2m, m2 at x=-4m, and m3 at x=0m. Draw a dot or a small circle at each of these positions and label them with their corresponding masses.

Step 2: Calculate total mass

To find the center of mass, we need to compute the total mass of the system, which is the sum of the individual masses: Total Mass = \(m_1 + m_2 + m_3\) Total Mass = \(8 kg + 4 kg + 1 kg = 13 kg\)

Step 3: Calculate the center of mass

Using the equation for the center of mass, we can now calculate the weighted average of the positions: $$\bar{x} = \frac{m_1x_1 + m_2x_2 + m_3x_3}{m_1 + m_2 + m_3}$$ Replace with given values: $$\bar{x} = \frac{(8 kg)(2 m) + (4 kg)(-4 m) + (1 kg)(0 m)}{13 kg}$$ Calculate the numerator: $$= \frac{16 kgm - 16 kgm + 0 kgm}{13 kg}$$ $$= \frac{0 kgm}{13 kg}$$ and then the quotient: $$\bar{x} = 0 m$$

Step 4: Plot the center of mass

The center of mass is at x=0m. On the number line, draw a vertical dashed line or arrow pointing towards the number line, label it with "center of mass" and indicate that its position is at x=0m. In conclusion, the center of mass for this system is located at x=0m on the number line.