Falling Box with Pulley | Physics with Professor Matt Anderson | M12-23

Introduction

In this tutorial, we will explore the dynamics of a falling box connected to a pulley. This scenario demonstrates key principles of physics, particularly the relationship between forces, acceleration, and motion. Understanding how to calculate the downward acceleration of the box provides insights into real-world applications, such as elevator systems and mechanical engineering.

Step 1: Understand the System Components

To analyze the falling box with a pulley, we need to identify the key components involved in the system.

• Box: The object that is falling under the influence of gravity.
• Pulley: A wheel that changes the direction of the tension force in the rope.
• Rope: Connects the box to the pulley and transmits forces.

Practical Tip

Make sure to visualize the setup. Draw a simple diagram showing the box, pulley, and rope to clarify the forces acting on each component.

Step 2: Identify the Forces Acting on the Box

Before calculating the acceleration, determine all the forces acting on the box.

• Weight (W): The force due to gravity, calculated as: [ W = m \cdot g ] where ( m ) is the mass of the box and ( g ) is the acceleration due to gravity (approximately ( 9.81 , m/s^2 )).

• Tension (T): The force in the rope pulling upward on the box.

Common Pitfall

Many assume the box will fall with an acceleration equal to ( g ). However, because of the tension in the rope, the actual acceleration will be less than ( g ).

Step 3: Apply Newton's Second Law

Use Newton's Second Law to set up the equation for the box:

• The net force acting on the box can be expressed as: [ F_{net} = W - T ]

• According to Newton’s Second Law: [ F_{net} = m \cdot a ] where ( a ) is the acceleration of the box.

Combine the Equations

Set the two expressions for ( F_{net} ) equal to each other: [ m \cdot a = m \cdot g - T ]

Step 4: Consider the Pulley’s Rotation

The tension in the rope not only affects the box but also causes the pulley to rotate.

• If the pulley has a moment of inertia ( I ) and a radius ( r ), the relationship between the tension and the angular acceleration ( \alpha ) of the pulley is given by: [ T \cdot r = I \cdot \alpha ]

• The angular acceleration is related to the linear acceleration of the box: [ \alpha = \frac{a}{r} ]

Step 5: Solve the Equations Simultaneously

Using the above relationships, you can substitute and solve for ( a ):

1. Substitute ( \alpha ) in terms of ( a ) into the tension equation.
2. Rearrange the equations to isolate ( a ).

Example Calculation

Assuming you have values for the mass of the box and the moment of inertia of the pulley, plug these into the equations to find ( a ).

Conclusion

In this tutorial, we explored how to analyze a falling box connected to a pulley by:

• Identifying the system components and forces.
• Applying Newton’s Second Law.
• Considering the effects of the pulley’s rotation.

By following these steps, you can calculate the downward acceleration of the box in various applications. As a next step, consider experimenting with different masses and pulley designs to see how they affect the system's dynamics.