Rolling Motion (Free Wheels): Videos & Practice Problems

Rolling motion involves rigid bodies that both rotate and translate, exemplified by a roll of toilet paper. The key relationship is between the center of mass velocity (v_cm) and angular velocity (ω), expressed as v=Rω. For a wheel, the velocity at the top is v = 2Rω, and at the bottom, it is always zero. These equations apply under the condition of rolling without slipping, crucial for understanding dynamics in rotational motion.

Rolling Motion (Free Wheels)

Rolling motion, often referred to as "free wheels," involves rigid bodies that not only rotate around their central axis but also translate sideways. This concept can be illustrated using a roll of toilet paper. When the roll is fixed in place, it can spin (with an angular velocity, denoted as \( \omega \)), but it does not move sideways, resulting in a center of mass velocity (\( v \)) of zero. In contrast, when the roll is allowed to roll on a surface, it exhibits both rotation and translation, meaning both \( \omega \) and \( v \) are non-zero.

The relationship between the angular velocity and the linear velocity of the center of mass for a wheel of radius \( R \) is given by the equation:

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

This equation highlights that the linear velocity of the center of mass is directly proportional to the angular velocity and the radius of the wheel. It is important to note that this relationship applies specifically when the wheel rolls without slipping, a condition essential for the equations governing rolling motion to hold true.

In addition to the center of mass velocity, there are specific velocities at different points on the wheel. The velocity at the top of the wheel is:

\( v_{\text{top}} = 2R \omega \)

And the velocity at the bottom of the wheel, relative to the floor, is:

\( v_{\text{bottom}} = 0 \)

These relationships can be summarized as follows: the velocity at the top is twice that of the center of mass, while the bottom point remains stationary relative to the ground.

For example, consider a wheel with a radius of 0.3 meters rolling without slipping at a speed of 10 meters per second. To find the angular speed (\( \omega \)), we can rearrange the equation for linear velocity:

\( \omega = \frac{v_{\text{cm}}}{R} = \frac{10 \, \text{m/s}}{0.3 \, \text{m}} = 33.33 \, \text{radians/second} \)

Furthermore, the speed of a point at the bottom of the wheel relative to the floor is always zero for a rolling wheel, regardless of its angular speed. Thus, in this scenario, while the angular speed is 33.33 radians per second, the bottom point's speed remains at zero.

Understanding these principles of rolling motion is crucial for solving problems related to dynamics and kinematics of rigid bodies in motion.

Speeds at points on a wheel

In this scenario, we analyze the motion of a car that accelerates from rest over a period of 10 seconds. The initial velocity of the car is 0 m/s, and the angular acceleration of the tires is given as \( \alpha = 8 \, \text{radians/second}^2 \). The radius of the tires is \( r = 0.4 \, \text{meters} \). Our goal is to determine the final angular speed of the tires after 10 seconds and the linear speeds at different points on the tire.

To find the final angular speed (\( \omega_{\text{final}} \)) of the tires, we can use the first equation of angular motion:

\( \omega_{\text{final}} = \omega_{\text{initial}} + \alpha t \)

Given that the initial angular speed (\( \omega_{\text{initial}} \)) is 0, we substitute the values:

\( \omega_{\text{final}} = 0 + (8 \, \text{radians/second}^2)(10 \, \text{seconds}) = 80 \, \text{radians/second} \)

Next, we calculate the linear speeds at the top, center, and bottom of the tire. The tire is in rolling motion, which allows us to apply the following relationships:

1. The speed at the top of the tire (\( v_{\text{top}} \)) is given by:

\( v_{\text{top}} = 2r\omega \)

2. The speed at the center of mass (\( v_{\text{cm}} \)) is:

\( v_{\text{cm}} = r\omega \)

3. The speed at the bottom of the tire (\( v_{\text{bottom}} \)) is always 0 due to the rolling motion.

Substituting the values into these equations:

\( v_{\text{top}} = 2(0.4 \, \text{meters})(80 \, \text{radians/second}) = 64 \, \text{meters/second} \)

\( v_{\text{cm}} = (0.4 \, \text{meters})(80 \, \text{radians/second}) = 32 \, \text{meters/second} \)

Thus, the final results are:

- The final angular speed of the tires is \( 80 \, \text{radians/second} \).

- The speed at the top of the tire is \( 64 \, \text{meters/second} \).

- The speed at the center of mass is \( 32 \, \text{meters/second} \).

- The speed at the bottom of the tire is \( 0 \, \text{meters/second} \).

This analysis illustrates the relationship between linear and angular motion, emphasizing the importance of understanding both concepts in the context of rolling objects.

A long, light rope is wrapped around a cylinder of radius 40 cm, which is at rest on a flat surface, free to move. You pull horizontally on the rope, so it unwinds at the top of the cylinder, causing it to begin to roll without slipping. You keep pulling until the cylinder reaches 10 RPM. Calculate the speed of the rope at the instant the cylinder reaches 10 RPM.

• A. 0.21 m/s
• B. 0.42 m/s
• C. 0.84 m/s
• D. 1.68 m/s

Answer: (C)

Here’s what students ask on this topic:

What is rolling motion and how does it differ from pure rotation?

Rolling motion involves a rigid body that both rotates around its own axis and translates along a surface. In contrast, pure rotation occurs when a body spins around a fixed axis without any translational movement. For example, a roll of toilet paper fixed in place can spin (pure rotation), but if it rolls on the floor, it both spins and moves sideways (rolling motion). The key relationship in rolling motion is between the center of mass velocity (v_cm) and angular velocity (ω), given by $v=R\omega$.

What is the condition for rolling motion without slipping?

The condition for rolling motion without slipping is that the point of contact between the rolling object and the surface must have zero velocity relative to the surface. This ensures that the object rolls smoothly without sliding. Mathematically, this condition is expressed as $v=R\omega$, where v is the velocity of the center of mass, R is the radius of the object, and ω is the angular velocity. This relationship ensures that the tangential speed at the point of contact matches the translational speed of the center of mass.

How do you calculate the angular speed of a rolling wheel?

To calculate the angular speed (ω) of a rolling wheel, you can use the relationship between the center of mass velocity (v_cm) and the radius (R) of the wheel. The formula is $\omega =\frac{v}{R}$. For example, if a wheel with a radius of 0.3 meters rolls with a center of mass velocity of 10 meters per second, the angular speed would be $10/0.3=33.3$ radians per second.

What are the velocities at different points on a rolling wheel?

For a rolling wheel, the velocities at different points vary. The velocity at the center of mass (v_cm) is given by $v=R\omega$. The velocity at the top of the wheel is twice the center of mass velocity, $2R\omega$, while the velocity at the bottom of the wheel is zero relative to the floor. These relationships hold true under the condition of rolling without slipping.

Why is the velocity at the bottom of a rolling wheel zero relative to the floor?

The velocity at the bottom of a rolling wheel is zero relative to the floor because, at the point of contact, the wheel's tangential speed due to rotation exactly cancels out its translational speed. This is a consequence of the rolling without slipping condition, where the point of contact must have no relative motion with respect to the surface. Mathematically, this is expressed as $v=R\omega$, ensuring that the tangential velocity at the bottom is zero.