How Do Polar Coordinates Explain a Bead's Velocity on a Rotating Wheel?

In summary, in the book 'Introduction To Mechanics' by Kleppner and Kolenkow, an example is given where a bead is moving along the spoke of a wheel at a constant speed u m/s. The wheel is rotating with uniform angular velocity dθ/dt = ω radians per second about an axis fixed in space. At t = 0, the spoke is along the x axis and the bead is at the origin. The velocity of the bead at time t in polar coordinates is ur + uωtθ, where the bead is at radius ut on the spoke. This equation is only valid until the bead reaches the rim of the wheel. This may have initially caused confusion because the direction of motion of the bead
  • #1
Precipitation
2
0
Note: All bold and underlined variables in this post are base vectors

I was reading the book 'Introduction To Mechanics' by Kleppner and Kolenkow and came across an example I don't quite understand. The example is this: a bead is moving along the spoke of a wheel at constant speed u m/s. The wheel rotates with uniform angular velocity dθ/dt = ω radians per second about an axis fixed in space.
At t = 0 the spoke is along the x axis, and the bead is at the origin. The book then says that the velocity of the bead at time t in polar coordinates is ur + uωtθ. Elaborating, the text says "at time t, the bead is at radius ut on the spoke."

What I don't understand is why u can be used in this calculation without any modification. If the bead is at radius ut at time t then the velocity would increase indefinitely and the spoke would have a position vector longer than the wheel it was attached to, which obviously doesn't make sense. Am I misunderstanding something about polar coordinates/vectors here or am I misunderstanding the example?
 
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  • #2
Precipitation said:
Note: All bold and underlined variables in this post are base vectors

I was reading the book 'Introduction To Mechanics' by Kleppner and Kolenkow and came across an example I don't quite understand. The example is this: a bead is moving along the spoke of a wheel at constant speed u m/s. The wheel rotates with uniform angular velocity dθ/dt = ω radians per second about an axis fixed in space.
At t = 0 the spoke is along the x axis, and the bead is at the origin. The book then says that the velocity of the bead at time t in polar coordinates is ur + uωtθ. Elaborating, the text says "at time t, the bead is at radius ut on the spoke."

What I don't understand is why u can be used in this calculation without any modification. If the bead is at radius ut at time t then the velocity would increase indefinitely and the spoke would have a position vector longer than the wheel it was attached to, which obviously doesn't make sense. Am I misunderstanding something about polar coordinates/vectors here or am I misunderstanding the example?

Obviously, eventually the bead will reach the rim of the wheel. That equation is only valid until then.
 
  • #3
PeroK said:
Obviously, eventually the bead will reach the rim of the wheel. That equation is only valid until then.

That makes sense. I was conceptualising it as the bead reversing direction as the wheel completed successive revolutions. Thanks for the help.
 

Related to How Do Polar Coordinates Explain a Bead's Velocity on a Rotating Wheel?

1. What are polar coordinates of a vector?

Polar coordinates of a vector represent its position or magnitude in terms of a distance (r) and angle (θ) from a fixed point or origin. It is an alternative way of representing a vector in a two-dimensional space.

2. How do you convert cartesian coordinates to polar coordinates of a vector?

To convert cartesian coordinates (x,y) to polar coordinates (r,θ), you can use the following equations: r = √(x² + y²) and θ = tan⁻¹(y/x). These equations are based on the Pythagorean theorem and trigonometric functions.

3. What is the significance of the angle in polar coordinates of a vector?

The angle (θ) in polar coordinates of a vector indicates the direction or orientation of the vector. It is measured counterclockwise from the positive x-axis to the vector.

4. Can polar coordinates of a vector be negative?

Yes, both the distance (r) and angle (θ) in polar coordinates of a vector can be negative. A negative distance represents a vector in the opposite direction, while a negative angle indicates a vector in the clockwise direction.

5. How do you add or subtract vectors in polar coordinates?

To add or subtract vectors in polar coordinates, you can use the parallelogram law or the triangle rule. In the parallelogram law, you draw two vectors from the same origin and complete the parallelogram. The diagonal of the parallelogram represents the sum or difference of the two vectors. In the triangle rule, you draw a triangle connecting the tips of the two vectors, and the third side of the triangle represents the sum or difference of the vectors.

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