Introductory Rotational Form of Newton's Second Law Problem

Summary

In this physics lesson, a problem is presented where a uniform solid disk rotates about a frictionless axle, with a force of 11 newtons applied via a string. The class works through the calculation of angular acceleration using the rotational inertia of the disk and the torque produced by the applied force. The discussion includes the necessary formulas and definitions, such as rotational inertia and torque, with a focus on understanding how to define positive torque directions. Ultimately, the students learn the importance of careful analysis and proper definitions when solving physics problems involving rotational motion.

Takeaways

• 😀 A uniform, solid disk rotates about a frictionless axle at its center of mass, mounted on a wall parallel to the plane of the disk.
• 😀 The force applied to the disk is 11 newtons, and a string with negligible mass wraps around the disk.
• 😀 The radius of the disk is 0.18 meters, and its mass is 1.5 kilograms.
• 😀 The angular acceleration of the disk is the main question to be solved.
• 😀 The rotational inertia of a solid disk about its center of mass is given by the formula 1/2MR².
• 😀 The rotational inertia of the disk in the problem is calculated as 0.0243 kg·m².
• 😀 The forces acting on the disk include the force normal, gravity, and the applied force.
• 😀 Torques are summed around the axis of rotation, with the positive direction defined as counterclockwise.
• 😀 The torque caused by the applied force is opposite to the positive direction and is clockwise.
• 😀 The angular acceleration is calculated using the formula α = -r * F / I, where r is the radius, F is the force, and I is the rotational inertia.
• 😀 The final result for the angular acceleration is -81 radians per second squared, indicating a clockwise rotation based on the chosen positive torque direction.

Q & A

What is the scenario described in the script?

A uniform, solid disk is mounted on a frictionless axle and rotated. A string of negligible mass wraps around the disk and is pulled by a force of 11 newtons. The problem involves calculating the angular acceleration of the disk.

What are the given values for the disk in the problem?

The radius of the disk is 0.18 meters, the mass of the disk is 1.5 kilograms, and the force applied to the string is 11 newtons.

What is the formula for rotational inertia of a solid disk?

The rotational inertia (moment of inertia) of a solid disk about its center of mass is given by the formula I = (1/2) * M * R^2, where M is the mass of the disk and R is its radius.

How was the rotational inertia calculated in the script?

The rotational inertia was calculated as (1/2) * 1.5 kg * (0.18 m)^2, which equals 0.0243 kg·m^2.

What is the concept of torque as explained in the script?

Torque is defined as the force applied multiplied by the distance from the axis of rotation, as well as the sine of the angle between the radius vector and the applied force. In this case, torque depends on the radius of the disk and the angle between the applied force and the radius.

Why do some forces not contribute to the net torque in this situation?

The forces acting on the axle and the center of mass do not contribute to the net torque because their distance (r value) from the axis of rotation is zero, leading to no torque.

What does the direction of positive torque refer to in the script?

In the script, the direction of positive torque is defined as counterclockwise, which is out of the board. The applied force causes a negative torque, meaning clockwise or into the board.

What assumption was made about the angle between the applied force and the radius?

The angle between the applied force and the radius is assumed to be 90 degrees because the string is tangent to the disk's surface, and any angle will result in the force being at a right angle to the radius.

How is angular acceleration calculated in the problem?

Angular acceleration is calculated by dividing the net torque by the rotational inertia. Since the torque is negative, the equation is angular acceleration = - (r * F) / I, resulting in an angular acceleration of -81.48 radians per second squared.

Why was the angular acceleration calculated as negative, and does it make sense?

The angular acceleration is negative because the torque is applied in a clockwise direction, which was chosen as negative based on the defined positive torque direction. While negative angular acceleration may seem odd, it correctly reflects the direction of rotation as defined in the problem setup.