Rolling Acceleration Down an Incline

Summary

In this video, Mr.p and his students explore the physics of a uniform solid cylinder rolling down an incline. They apply principles of mechanical energy conservation, rotational inertia, and forces acting on the cylinder to determine its acceleration. Through a series of engaging discussions and calculations, the class concludes that the cylinder's acceleration depends on the incline angle and the gravitational force, but not its mass or radius. The final acceleration is experimentally validated, matching theoretical predictions. The students wrap up with a lively, positive reflection on the results, emphasizing how the physics works in real-world applications.

Takeaways

• 😀 The problem involves determining the acceleration of a uniform solid cylinder rolling without slipping down an incline with an angle θ.
• 😀 The rotational inertia of a uniform solid cylinder is given as (1/2) * mass * radius squared.
• 😀 Mechanical energy is conserved as the cylinder rolls down the incline, since no external work is done on the system.
• 😀 The cylinder starts from rest, meaning its initial kinetic energy is zero.
• 😀 The initial mechanical energy of the cylinder consists of gravitational potential energy, and there are no elastic potential energies involved.
• 😀 At the final point, the cylinder has both translational and rotational kinetic energies, which are used in the conservation of energy equation.
• 😀 The velocity of the center of mass of a rolling object without slipping is related to its angular velocity through the equation: velocity final = radius * angular velocity final.
• 😀 By substituting angular velocity terms into the energy equation, the final velocity squared is expressed as (4/3) * gravity * height initial.
• 😀 The displacement along the incline is related to the initial height and the sine of the incline angle, using trigonometric principles.
• 😀 The acceleration of the cylinder down the incline is independent of its mass and radius, and depends only on the incline angle and the acceleration due to gravity.
• 😀 The calculated acceleration of the cylinder down the incline is confirmed by measured data, with both the theoretical and experimental values being 1.73 m/s².

Q & A

What is the problem that Bobby and Bo are trying to solve?

They are trying to determine the acceleration of a uniform solid cylinder rolling without slipping down an incline, with a given incline angle θ.

What is the equation for the rotational inertia of a uniform solid cylinder?

The rotational inertia of a uniform solid cylinder about its long cylindrical axis is given by one-half times the mass times the radius squared.

Why is mechanical energy considered conserved in this scenario?

Mechanical energy is conserved because there is no external force doing work to add or remove energy from the system, and the force of static friction does not do work since there is no sliding between the cylinder and the incline.

What type of energy does the system have initially and finally?

Initially, the system has only gravitational potential energy. Finally, the system has both translational kinetic energy (due to the linear motion of the center of mass) and rotational kinetic energy (due to the cylinder's rotation).

How is the final velocity of the cylinder related to its angular velocity?

The final velocity of the cylinder is related to its angular velocity through the equation: velocity final = cylinder radius times angular velocity final.

How is the final velocity squared calculated in the energy conservation equation?

The final velocity squared is calculated by substituting the expression for angular velocity into the conservation of energy equation, resulting in velocity final squared = (4/3) times acceleration due to gravity times height initial.

How does the displacement along the incline relate to the height of the cylinder?

The height of the cylinder is related to the displacement along the incline through the equation: height initial = displacement parallel times the sine of the incline angle.

What key factors affect the acceleration of the cylinder rolling down the incline?

The key factors affecting the acceleration are the acceleration due to gravity, the incline angle θ, and the shape of the object, as determined by the factor in the rotational inertia equation.

What conclusion is reached about the mass and radius of the cylinder in the final acceleration equation?

The mass and radius of the cylinder cancel out in the final equation for acceleration, meaning they do not affect the acceleration of the cylinder rolling without slipping down the incline.

How was the acceleration of the cylinder measured, and what was the result?

The acceleration of the cylinder was measured by determining the slope of the velocity versus time curve. The measured acceleration was 1.73 m/s², which matched the predicted value.