Rolling without Slipping Comment: RPM (rotations-per-minute) and MPH (miles-per-

Question

Rolling without Slipping Comment: RPM (rotations-per-minute) and MPH (miles-per-hour) are still used quite commonly in many industries so practicing engineers need to be able to readily convert to rad/sec and Km /sec, etc. as necessary. Also, you might see these units on exams on occasion so you might want to have the conversions handy. Assume you can idealize an automotive wheel as a circular rigid body with radius R = 0.5 m that is rolling without slipping at it's point of contact C. Points A, B, C and D are on the periphery of the wheel and point O is at it's center as shown. You establish an XY coordinate system as shown that coincides with the point of contact at C at the instant of interest. If the wheel is rotating at omega = 256 RPM CW : How many MPH is the vehicle traveling and in which direction? How many m/s is this? Write a general XY velocity vector for pt O in terms of R & omega. (note remember that positive rotation is always CCW in the XY plane according to the RH rule). Write a general equation for the velocity vector of any point on the periphery of the wheel in terms of R & co. Evaluate your equation for the angular velocity specified and predict, in m/s, the velocities of points A, B & C. Draw a circle to represent the wheel and show the velocity at these points along with the velocity of pt O in the appropriate direction and approximately to scale relative to one another. Where is the Instant Center at this moment in the XY system. Is the instant center moving ? If so, describe it's motion in the XY system.

Explanation / Answer

a) v=wr

w=256RPM=256*2 pi* 60 radians per hour

r=0.5m=0.5*0.000621371 Mile=0.000310686 mile

v=256*2 pi* 60*0.000310686 mile per hour = 30 MPS.. negative x direction,

=13.4 m/s

b) Vx= - wR i.....because if w is in CCW, v goes towards -x, i is unit vector in x direction

Vy=0....height of wheel is fixed

c) v= w x R..... it is cross product of w and R.......this is velocity respect to center.

v= w x R - wR i. for general frame...after adding velocity of center

d)Va=w x R   - wR i = wR j - wRi ........top left direction (have both y and -x component)

Vb=w x R   - wR i = -2wRi ....left direction or negative x direction

Vc=wR i - wRi = 0 it is zero so no direction

e)at center

f)yes moving, in negative x direction

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