a wheel of radius 3m rolls at 17 rads/s. how fast is a point on the rim of wheel rising when the point is pi/3 radians above the horizontal(and rising)?

Question

A wheel with radius 3m rolls at 17 rads/s. How fast is a point on the rim of wheel rising when the point is pi/3 radians above the horizontal (and rising)?

Answer 1

The speed at which a point on the rim travels is equal to the angular velocity (17 radians per second, 17 sec^-1) times the radius (3m).  So the speed of the point is 51 m/s.

 

However, as the point goes around the circle formed by the wheel, it's direction changes.  So, for example, when the point is at an angle of 0 above the horizontal, the point's direction of motion is straight up, so the upward (rising) velocity is equal to the speed of the point.  However, when the point is at an angle of π/2, the entire motion is horizontal and the point is not rising (or falling) at all.

 

The component of the velocity that is pointing in the up-down direction is equal to the total speed times the cosine of the angle (above horizontal).  cos π/3 = 0.5, so at that angle the point is rising at a rate of 0.5*51 = 25.5 m/s.