Question

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

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

Step-by-step solution

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 \(m_1\).

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: \(T_3 - m_3g = m_3a\).

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 \(m_3\).

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 \(a\): \(a = \frac{m_3g - m_2g - m_1g}{m_1}\)

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.

Key concepts

Gravitational Force

Gravitational force is a fundamental concept that plays a pivotal role in understanding motion and interaction of bodies under gravity. It is the attractive force exerted by any mass towards the center of the Earth. This force acts on all objects with mass, pulling them downward. For any object, you can calculate gravitational force using the formula: \[ F_g = mg \]where:

• \( F_g \) represents the gravitational force
• \( m \) signifies the mass of the object
• \( g \) is the acceleration due to gravity, approximately \( 9.81 \ m/s^2 \) on Earth

In the given exercise, each mass, whether it is \( m_1, m_2, \) or \( m_3 \), experiences a gravitational pull downward with its own distinct amount of force based on its mass. This downward force must be counteracted by the tension in the pulleys for the system to be in equilibrium.

Tension Force

Tension force is crucial when dealing with objects connected by strings or ropes over pulleys. It refers to the force that is transmitted through a string, cable, or rope when it is pulled tight by forces acting from opposite ends. This force acts parallel to the string in the opposite direction to gravitational pull, aiming to balance the system.In the exercise, the tension force \( T \) is the counteracting force for each of the hanging masses. For each mass, the rope provides a tension that opposes the weight, creating a balance necessary for calculating acceleration:

• For \( m_1 \), tension \( T_1 \) must balance against \( m_1g \) and provide the net force leading to the calculated acceleration.
• Similarly, for \( m_2 \) and \( m_3 \), tensions \( T_2 \) and \( T_3 \) relate similarly counteracting their respective weights.

The relationship and difference between these tensions play a key role in resolving the net force and hence allowing the computation of acceleration.

Acceleration

Acceleration is the rate of change of velocity of an object. Newton's second law defines it as directly proportional to the net force acting on an object and inversely proportional to its mass. Mathematically, this relationship is expressed as:\[ F_{net} = ma \]where:

• \( F_{net} \) is the net force acting on the object
• \( m \) is the mass
• \( a \) stands for acceleration

In the scenario given, knowing the masses and gravitational force, applying Newton's second law helps us pinpoint the acceleration of \( m_1 \). The exercise involves calculating the net forces using the relationship between the gravitational and tension forces. The net force equation:\[T_3 - T_2 - m_1g = m_1a\]is rearranged to solve for \( a \), revealing that the calculated acceleration is negative, meaning that \( m_1 \) is actually accelerating in the upward direction away from gravity—not a very intuitive result, but absolutely crucial in understanding the system's behavior.