Question

Particle A is at the origin and has a mass of \(30.0 \mathrm{g} .\) Particle B has a mass of \(10.0 \mathrm{g} .\) Where must particle \(\mathrm{B}\) be located if the coordinates of the \(\mathrm{cm}\) are \((x, y)=(2.0 \mathrm{cm},\) $5.0 \mathrm{cm}) ?$

Short answer

Answer: The coordinates of Particle B are (8.0 cm, 20.0 cm).

Step-by-step solution

Write down the formula for the center of mass in two dimensions

The formula for the center of mass in two dimensions is given by: \(x_{cm} = \frac{m_Ax_A + m_Bx_B}{m_A + m_B}\) \(y_{cm} = \frac{m_Ay_A + m_By_B}{m_A + m_B}\) where \(m_A\) and \(m_B\) are the masses of particles A and B, respectively, and \((x_A, y_A)\) and \((x_B, y_B)\) are the coordinates of particles A and B, respectively.

Substitute the given values into the x-coordinate formula for center of mass

We are given \(x_{cm} = 2.0 cm\), \(m_A = 30.0 g\), \(x_A = 0\), and \(m_B = 10.0 g\). Plugging these values into the x-coordinate formula, we get: \(2.0 = \frac{30.0 \times 0 + 10.0 \times x_B}{30.0 + 10.0}\)

Solve for the x-coordinate of Particle B

Simplifying and solving for \(x_B\), we obtain: \(2.0 = \frac{10.0x_B}{40.0}\) \(x_B = \frac{2.0 \times 40.0}{10.0}\) \(x_B = 8.0 \;\mathrm{cm}\)

Substitute the given values into the y-coordinate formula for center of mass

We are given \(y_{cm} = 5.0 cm\), \(m_A = 30.0 g\), \(y_A = 0\), and \(m_B = 10.0 g\). Plugging these values into the y-coordinate formula, we get: \(5.0 = \frac{30.0 \times 0 + 10.0 \times y_B}{30.0 + 10.0}\)

Solve for the y-coordinate of Particle B

Simplifying and solving for \(y_B\), we obtain: \(5.0 = \frac{10.0y_B}{40.0}\) \(y_B = \frac{5.0 \times 40.0}{10.0}\) \(y_B = 20.0 \;\mathrm{cm}\)

Combine x and y coordinates for Particle B

Now that we have found both the x and y coordinates of Particle B, we can combine them to find its location: The location of Particle B is \((8.0 \;\mathrm{cm}, 20.0 \;\mathrm{cm})\).