# Dynamics Question with Rotational Axis Theorem

• ryank614
In summary, a cyclist rides a circular track with a radius of 30m at a constant speed of 10m/s. The bicycle is banked at 15 degrees inward from the vertical. To find the acceleration of a tack with a radius of .4m as it passes through the highest point of its path, the angular velocity and angular acceleration must be calculated. The velocity and acceleration of the tack with respect to the point of contact of the wheel on the track must also be determined, and then all of the knowns can be plugged into the acceleration equation.
ryank614

## Homework Statement

A cyclist rides around a circular track (R= 30m) such that the point of contact of the wheel on the track moves at a constant speed of 10m/s. The bicycle is banked at 15 degrees inward from the vertical. Find the acceleration of a tack in the tire (.4m radius) as it passed through the highest point of its path. Use cylindrical unit vectors in expressing the answer

## Homework Equations

acceleration = $$\ddot{P}$$ + $$\alpha$$ x P + 2 $$\omega$$ x $$\dot{P}$$ + $$\omega$$ x ($$\omega$$ x P)

Where the acceleration of the tack = acceleration of tack with respect to the second frame (the point of contact of the wheel) + angular acceleration cross product with vector from center of circular track to tack + 2 * angular velocity cross product with velocity of the tack with respect to the point of contact of wheel + $$\omega$$ x ($$\omega$$ x P)

## The Attempt at a Solution

I know this problem is a matter of finding all the unknowns. I have found $$\omega$$ and angular acceleration.

I think I can also find OP (we will call O the center of the track, so this is the vector to the tack).

How do you find the velocity and acceleration of the tack with respect to the point of contact of the wheel on track?

Once I have all the knowns, I can plug it into the acceleration equation. \omega = \frac{V}{R} = \frac{10}{30} = 0.333 rad/sAlpha = \frac{\omega^{2}}{R} = (0.333)^2/30 = 0.00555 rad/s^2OP = \sqrt{R^2 - (0.4)^2} = 29.6m Velocity of tack with respect to point of contact of wheel on track = ? Acceleration of tack with respect to point of contact of wheel on track = ? Then, acceleration of tack = \ddot{P} + \alpha x P + 2 \omega x \dot{P} + \omega x (\omega x P) = ?

## 1. What is the rotational axis theorem?

The rotational axis theorem states that the angular momentum of a system is conserved when there is no external torque acting on the system. This means that the total angular momentum of a system remains constant, regardless of any internal forces or rotations within the system.

## 2. How is the rotational axis theorem applied in dynamics?

In dynamics, the rotational axis theorem is often used to analyze the motion of rotating objects or systems. It allows us to determine the final angular velocity of a system after a rotational force is applied, as well as the changes in angular velocity throughout the motion.

## 3. Can the rotational axis theorem be applied to all types of motion?

No, the rotational axis theorem only applies to rotational motion, where objects or systems are rotating around a fixed axis. It does not apply to linear motion, where objects are moving in a straight line.

## 4. What are some real-life applications of the rotational axis theorem?

The rotational axis theorem is used in a variety of fields, including engineering, physics, and sports. It is used to design and analyze the motion of rotating objects, such as gears, wheels, and turbines. It is also used in sports, such as figure skating and gymnastics, to understand and improve the performance of rotational movements.

## 5. Is the rotational axis theorem always true?

No, the rotational axis theorem is a simplified model that assumes there are no external torques acting on a system. In reality, there are often external forces or torques that can affect the angular momentum of a system. However, the theorem is a useful approximation in many cases and can help us understand the general behavior of rotating systems.

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