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birulami
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Let a bicycle run straight on a plane past you from left to right with constant speed. As a representative point of the bicycle take the center of the back axle. Since the bicycle has constant speed with regard to the plane, so has the axle (trivial). In addition a point on the rim of the wheel, say R, has constant angular velocity. Even more, with regard to the axle center, the absolute value of the velocity of R, i.e. [itex]|v_R|[/itex] is constant.
But, with regard to the plane, the velocity of R is not constant. As an example let's look at the horizontal speed of R between the angles 12 o'clock (top) and 3 o'clock (right). R moves [itex]2\pi r/4+r[/itex] if the radius of the wheel is r. Why? At the start, R is on top of the axle, at the end, the axle moved a quarter circumference and R is a distance of r in front of it. With a similar argument we find that between angles from 3 to 6 o'clock, R only moves [itex]2\pi r/4-r[/itex]. Since the time for both moves is the same, the horizontal speed with regard to the plane is obviously different. In both cases the vertical distance traveled by R is r. Consequently the absolute value of the velocity vector of R with regard to the plane is not constant.
Now the questions:
Every hint appreciated,
(No, not a homework assignment )
But, with regard to the plane, the velocity of R is not constant. As an example let's look at the horizontal speed of R between the angles 12 o'clock (top) and 3 o'clock (right). R moves [itex]2\pi r/4+r[/itex] if the radius of the wheel is r. Why? At the start, R is on top of the axle, at the end, the axle moved a quarter circumference and R is a distance of r in front of it. With a similar argument we find that between angles from 3 to 6 o'clock, R only moves [itex]2\pi r/4-r[/itex]. Since the time for both moves is the same, the horizontal speed with regard to the plane is obviously different. In both cases the vertical distance traveled by R is r. Consequently the absolute value of the velocity vector of R with regard to the plane is not constant.
Now the questions:
- Is it possible to let the bicycle move such that the absolute value of the velocity of R with regard to the plane is constant? How does the mathematical function look like and how does the bicycle move then?
- Can we have constant speed of the bicycle as well as a constant absolute value of the velocity of R at the same time, if necessary drop the requirement of circular motion of R?
Every hint appreciated,
Harald.
(No, not a homework assignment )