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Chapter 4: Problem 47

Three objects with masses \(m_{1}=36.5 \mathrm{~kg}, m_{2} 19.2 \mathrm{~kg},\) and \(m_{3}=12.5 \mathrm{~kg}\) are hanging from ropes that run over pulleys. What is the acceleration of \(m_{1} ?\)

Short Answer

Expert verified
Answer: The acceleration of mass \(m_1\) is approximately \(-2.59 m/s^2\) in the upward direction.

Step by step solution

01

Identify forces acting on the masses

Each mass is acted upon by two forces: 1. The gravitational force (weight) which pulls the mass downward with a force \(F_g = mg\), where \(m\) is the mass and \(g\) is the gravitational acceleration (\(9.81 m/s^2\) in the downward direction). 2. The tension force (\(T\)) in the rope, which acts upward and counteracts the gravitational force. The net force acting on each mass will be a combination of these two forces, and we will use that to find the acceleration of \(m_1\).
02

Apply Newton's second law to each mass

For each mass, we write Newton's second law equation based on the net force acting on the mass. Let's denote the acceleration of \(m_1\) as \(a\). For \(m_1\), the net force is the difference between the tension force \(T_1\) and the gravitational force: \(T_1 - m_1g = m_1a\). For \(m_2\), the net force is the difference between the tension force \(T_2\) and the gravitational force: \(T_2 - m_2g = m_2a\). For \(m_3\), the net force is the difference between the tension force \(T_3\) and the gravitational force: \(T_3 - m_3g = m_3a\).
03

Analyze the tension forces

The tension forces are created by the three masses pulling on the ropes. We have one equation that relates all three tension forces: \(T_3 - T_2 = T_1\). This is because mass \(m_1\) needs to balance out the combined force from masses \(m_2\) and \(m_3\).
04

Solve for acceleration \(a\)

From Step 3, we can write \(T_1 = T_3 - T_2\). Now plug this into the equation for \(m_1\) from Step 2: \(T_3 - T_2 - m_1g = m_1a\) We want to eliminate \(T_2\) and \(T_3\) from the equation and solve for \(a\). We can plug the equations for \(m_2\) and \(m_3\) from Step 2 into the above equation: \((T_3 - m_3g) - (T_2 - m_2g) - m_1g = m_1a\) Solve this equation for \(a\): \(a = \frac{m_3g - m_2g - m_1g}{m_1}\)
05

Calculate the acceleration

Now plug in the given values for masses and gravitational acceleration into the equation and solve for the acceleration \(a\): \(a = \frac{12.5 * 9.81 - 19.2 * 9.81 - 36.5 * 9.81}{36.5} = -2.59~m/s^2\) The acceleration of mass \(m_1\) is approximately \(-2.59 m/s^2\). The negative sign indicates that the acceleration is in the opposite direction of the gravitational force, so mass \(m_1\) is moving upward.

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