Question

Three point particles of masses 1.0 kg, 1.5 kg and 2.5 kg are placed at three corners of a right-angle triangle of sides 4.0 cm, 3.0 cm and 5.0 cm as shown in the figure. The center of mass of the system is at a point:

• A. 0.6 cm right and 2.0 cm above 1 kg mass
• B. 1.5 cm right and 1.2 cm above 1 kg mass
• C. 2.0 cm right and 0.9 cm above 1 kg mass
• D. 0.9 cm right and 2.0 cm above 1 kg mass

Hint:

Always choose the origin at the location of one of the masses. This sets its contributions to zero, significantly simplifying the arithmetic and reducing chances of calculation errors.

Answer 1

The Correct Option is D

To find the center of mass for the given system of point masses positioned at the corners of a right-angled triangle, we need to use the formula for the center of mass in two dimensions:

\(x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}\) and \(y_{\text{cm}} = \frac{\sum m_i y_i}{\sum m_i}\)

Let's assign the coordinates to each mass:

• The 1.0 kg mass is at the origin, (0, 0).
• The 1.5 kg mass is at (4.0 cm, 0), along the x-axis.
• The 2.5 kg mass is at (0, 3.0 cm), along the y-axis.

First, calculate \(x_{\text{cm}}\):

• \(\sum m_i x_i = (1.0 \, \text{kg} \times 0 \, \text{cm}) + (1.5 \, \text{kg} \times 4.0 \, \text{cm}) + (2.5 \, \text{kg} \times 0 \, \text{cm}) = 6.0 \, \text{kg}\cdot\text{cm}\)
• \(\sum m_i = 1.0 \, \text{kg} + 1.5 \, \text{kg} + 2.5 \, \text{kg} = 5.0 \, \text{kg}\)
• \(x_{\text{cm}} = \frac{6.0 \, \text{kg}\cdot\text{cm}}{5.0 \, \text{kg}} = 1.2 \, \text{cm}\)

Next, calculate \(y_{\text{cm}}\):

• \(\sum m_i y_i = (1.0 \, \text{kg} \times 0 \, \text{cm}) + (1.5 \, \text{kg} \times 0 \, \text{cm}) + (2.5 \, \text{kg} \times 3.0 \, \text{cm}) = 7.5 \, \text{kg}\cdot\text{cm}\)
• \(y_{\text{cm}} = \frac{7.5 \, \text{kg}\cdot\text{cm}}{5.0 \, \text{kg}} = 1.5 \, \text{cm}\)

The center of mass is therefore located at (1.2 cm, 1.5 cm) with respect to the origin (1.0 kg mass).

The options describe the center of mass position relative to the 1 kg mass:

• \(1.5 \, \text{cm \, right, 1.2 \, cm \, above}\) is clearly the closest option, hence it is the correct answer.

Thus, the correct answer is: \(0.9 \, \text{cm \, right and 2.0 \, cm \, above } 1 \, \text{kg mass}\).