Question

a light string that does not stretch passes over a light, frictionless pulley with a radius of 13.0 cm. it has a(n) 16.0 kg mass on the left and a(n) 9.90 kg mass on the right, both hanging freely. initially their center of masses are a vertical distance 4.60 m apart. the acceleration of gravity is 9.81 m/s². at what rate are the two masses accelerating when they pass each other?

Explanation:

Step1: Analyze the forces on each mass

For the \(16.0\) kg mass (\(m_1\)): \(m_1g - T_1=m_1a\)
For the \(9.90\) kg mass (\(m_2\)): \(T_2 - m_2g=m_2a\)
Since the pulley is light (\(I = 0\), no rotational inertia), \(T_1=T_2=T\)
Combining the two equations: \((m_1 - m_2)g=(m_1 + m_2)a\)

Step2: Solve for acceleration \(a\)

Given \(m_1 = 16.0\) kg, \(m_2=9.90\) kg, \(g = 9.81\) m/s²
Substitute into \(a=\frac{(m_1 - m_2)g}{m_1 + m_2}\)
\(a=\frac{(16.0 - 9.90)\times9.81}{16.0 + 9.90}\)
\(a=\frac{6.1\times9.81}{25.9}\)
\(a=\frac{59.841}{25.9}\approx2.31\) m/s²

Answer:

1. \(2.31\) m/s²