Rotational Dynamics of Rolling Motion: Videos & Practice Problems

Rolling motion involves both linear and rotational dynamics, characterized by torque and angular acceleration. The relationship between linear velocity (v_cm) and angular velocity (ω) is given by vcm = rω. Static friction generates torque, essential for angular acceleration (α). For a solid cylinder rolling down an incline, the acceleration (a) can be derived as a = 2/3gsin(θ), and angular acceleration is α = ar.

Rotational Dynamics of Rolling Motion

In the study of rotational dynamics, particularly in rolling motion, we explore how objects rotate around an axis while also translating. This concept can be visualized with a rolling object, such as a toilet paper roll, which moves sideways while rotating. When analyzing such motion, it is essential to recognize that the object experiences both linear and angular acceleration.

For a rolling object, the relationship between linear velocity at the center of mass (\(v_{cm}\)) and angular velocity (\(\omega\)) is given by the equation:

\(v_{cm} = R \omega\)

where \(R\) is the radius of the object. Similarly, the linear acceleration (\(a\)) is related to angular acceleration (\(\alpha\)) by:

\(a = R \alpha\)

In rolling motion, static friction plays a crucial role. It is the force that provides the necessary torque to cause angular acceleration. The torque (\(\tau\)) can be expressed as:

\(\tau = F_{friction} \cdot R\)

where \(F_{friction}\) is the static friction force acting at the radius \(R\). This torque leads to angular acceleration, and it is important to note that if there is no static friction, there can be no angular acceleration. This principle is often summarized with the phrase, "You need static friction to have angular acceleration."

When analyzing a solid cylinder rolling down an inclined plane without slipping, we can derive the angular acceleration (\(\alpha\)). The forces acting on the cylinder include gravitational force (\(mg\)), which can be decomposed into components parallel and perpendicular to the incline. The net force acting down the incline is given by:

\(F_{net} = mg \sin(\theta) - F_{friction}\

where \(\theta\) is the angle of the incline. The torque due to static friction is the only torque acting on the cylinder, leading to the equation:

\(\tau = I \alpha\

where \(I\) is the moment of inertia of the cylinder, given by:

\(I = \frac{1}{2} m R^2\

By substituting the expressions for torque and angular acceleration, we can relate linear and angular quantities. The final expression for the linear acceleration (\(a\)) of the cylinder down the incline is:

\(a = \frac{2}{3} g \sin(\theta)\

From this, we can find the angular acceleration (\(\alpha\)) using the relationship between linear and angular acceleration:

\(\alpha = \frac{a}{R} = \frac{2g \sin(\theta)}{3R}\

This analysis highlights the interconnectedness of linear and rotational motion, emphasizing the importance of static friction in rolling dynamics. Understanding these principles is crucial for solving problems related to rolling objects in physics.

A hollow sphere 10 kg in mass and 2 m in radius rolls without slipping along a horizontal surface with 20 m/s. It then reaches an inclined plane that makes 37° with the horizontal, as shown. If it rolls up the incline without slipping, how long will it take to reach its maximum height? (Hint:You will need to first calculate its acceleration)

• A. 3.58 s
• B. 4.26 s
• C. 4.75 s
• D. 5.67 s

Answer: (D)

Here’s what students ask on this topic:

What is the relationship between linear velocity and angular velocity in rolling motion?

In rolling motion, the relationship between linear velocity (v_cm) and angular velocity (ω) is given by the equation:

$v_{\mathrm{cm}}=r\omega$

Here, v_cm is the linear velocity of the center of mass, r is the radius of the rolling object, and ω is the angular velocity. This equation indicates that the linear velocity of the center of mass is directly proportional to the angular velocity, with the radius as the proportionality constant. This relationship is crucial for understanding how rotational and linear motions are interconnected in rolling objects.

How does static friction contribute to rolling motion?

Static friction is essential for rolling motion because it generates the torque needed for angular acceleration (α). When an object rolls without slipping, static friction acts at the point of contact between the object and the surface. This frictional force creates a torque that causes the object to rotate. Without static friction, the object would either slide without rolling or not accelerate rotationally. The torque (τ) due to static friction (f_s) is given by:

$\tau =f_{s}r$

where r is the radius of the object. This torque results in angular acceleration, linking linear and rotational dynamics in rolling motion.

What is the formula for the acceleration of a solid cylinder rolling down an incline?

The acceleration (a) of a solid cylinder rolling down an incline without slipping can be derived using Newton's second law for both linear and rotational motion. The formula is:

$a=\frac{2g\sin \left(\theta \right)}{3}$

Here, g is the acceleration due to gravity, and θ is the angle of the incline. This equation shows that the acceleration of the cylinder depends on the gravitational force component along the incline and is reduced by a factor of 2/3 due to the rotational inertia of the cylinder.

How do you derive the angular acceleration of a rolling object?

To derive the angular acceleration (α) of a rolling object, you can use the relationship between linear acceleration (a) and angular acceleration. For a rolling object, this relationship is given by:

$\alpha =\frac{a}{r}$

where r is the radius of the object. For example, for a solid cylinder rolling down an incline, if the linear acceleration is:

$a=\frac{2g\sin \left(\theta \right)}{3}$

then the angular acceleration is:

$\alpha =\frac{2g\sin \left(\theta \right)}{3r}$

This shows how the linear and angular accelerations are related through the radius of the rolling object.

What are the key equations used in solving rolling motion problems?

In solving rolling motion problems, several key equations are used to relate linear and rotational dynamics:

1. Newton's second law for linear motion: $\sum F=ma$

2. Newton's second law for rotational motion: $\sum \tau =I\alpha$

3. Relationship between linear and angular velocity: $v_{\mathrm{cm}}=r\omega$

4. Relationship between linear and angular acceleration: $a=r\alpha$

5. Torque due to static friction: $\tau =f_{s}r$

These equations help analyze the forces and torques acting on a rolling object, allowing you to solve for unknown quantities like acceleration and angular acceleration.