Question

you are trying to balance a tray of dishes to make it easy to carry. at one end of the 0.6 m tray are four plates with total mass 0.7 kg with their centers of mass exactly above the left edge of the tray. how far (in meters) from the left edge should you place a 1.9 kg jug of milk so that the centre of mass of the system is at the centre of the tray?

Explanation:

Step1: Set up the center - of - mass formula

The center - of - mass formula for a one - dimensional system is \(x_{cm}=\frac{m_1x_1 + m_2x_2}{m_1 + m_2}\). Let the left - edge of the tray be \(x = 0\). The length of the tray \(L=0.6m\), so the center of the tray \(x_{cm}=0.3m\). The mass of the plates \(m_1 = 0.7kg\) and \(x_1 = 0\). The mass of the jug \(m_2=1.9kg\) and let its position be \(x_2\).

Step2: Substitute values into the formula

Substitute into \(x_{cm}=\frac{m_1x_1 + m_2x_2}{m_1 + m_2}\), we get \(0.3=\frac{0.7\times0+1.9\times x_2}{0.7 + 1.9}\).

Step3: Solve for \(x_2\)

First, simplify the right - hand side of the equation: \(\frac{1.9x_2}{2.6}=0.3\). Then, solve for \(x_2\) by multiplying both sides by \(2.6\) and dividing by \(1.9\). \(x_2=\frac{0.3\times2.6}{1.9}=\frac{0.78}{1.9}\approx0.41m\)

Answer:

\(0.41m\)