a track is mounted on a large wheel that is free to turn with negligible friction about a vertical axis. a toy train of mass m is placed on the track and, with the system initially at rest, the electrical power is turned on. the train reaches a steady speed v with respect to the track. what is the angular speed of wheel if its mass is M and its radius is R? solve using variables
If angular momentum is conserved, then the sum of the train anglular momentum must be equal and opposite to the wheel. The speed v must be the train speed v'and the tangential speed v" of the wheel.
Using v', and v", you can determine the angular speed of the wheel (remember you have R, m, and M). Assume the wheel to be either a hoop, or a disk. State which, the problem is not specific on this. I would go for the disk.
thanks a bunch
To solve this problem, we need to apply the principles of conservation of energy and Newton's second law of motion.
First, let's define the variables:
- Mass of the toy train: m
- Mass of the wheel: M
- Radius of the wheel: R
- Steady speed of the train with respect to the track: v
- Angular speed of the wheel: ω (we want to find this)
Now, let's break it down step by step:
1. Conservation of Mechanical Energy:
We can assume that the system is isolated, so the total mechanical energy of the system remains constant. Initially, the system is at rest, so the initial mechanical energy is zero. At the steady speed v, the final mechanical energy is the kinetic energy of the train.
The initial mechanical energy is zero because there is no motion:
Initial Mechanical Energy = 0
The final mechanical energy is the kinetic energy of the train:
Final Mechanical Energy = (1/2) * m * v^2
2. Newton's Second Law of Motion:
The net force acting on the train is provided by the friction between the train and the wheel. According to Newton's second law, the net force is equal to the product of mass and acceleration. In this case, acceleration is the centripetal acceleration, since the train moves in a circular path.
The net force is equal to the product of the mass of the train and its centripetal acceleration:
Net Force = m * v^2 / R
3. Equating Mechanical Energy and Net Force:
Since the system is isolated, the final mechanical energy should be equal to the work done by the net force.
Final Mechanical Energy = Work done by net force
Substituting the values:
(1/2) * m * v^2 = m * v^2 / R * 2π
4. Solving for Angular Speed:
Using the above equation, we can solve for the angular speed ω of the wheel.
First, simplify the equation by canceling out common terms and rearranging:
(1/2) * v^2 = v^2 / R * 2π
Multiply both sides by R:
(1/2) * v^2 * R = v^2 * 2π
Divide both sides by v^2:
(1/2) * R = 2π
Multiply both sides by 2:
R = 4π
Finally, divide both sides by 2π:
R/2π = 2π/2π
Simplifying, we find:
R/2π = 1
Therefore, the angular speed of the wheel is:
ω = 1 rad/s
So, the angular speed of the wheel is 1 radian per second.