Examine the diagram. A piston rod, PQ, is
connected to a wheel at P and to a piston
at Q. As P moves around the wheel in a
counterclockwise direction, Q slides back
and forth.
there's a unit circle. origin to intersection at (1,0) is point P. Extending beyond the edges of the circle is a point Q that is connected to point P.
b)
What distance will Q have moved 1 s
after start-up? Give your answer to the
nearest hundredth of a unit
when the wheel has turned through an angle θ, P is at (x,y)=(cosθ,sinθ).
So, since the x-coordinate is at cosθ, Q has moved 1-cosθ left from its starting point.
You don't say how fast the wheel is turning, so there's no way for me to know how far it has turned in 1 second.
To determine the distance Q will have moved 1 second after start-up, we need to consider the motion of point P around the unit circle.
In one complete revolution around the unit circle, the circumference of the circle is 2π units. Therefore, in 1 second, point P will move a distance of 2π units.
Since point Q is connected to point P and slides back and forth, it will also move the same distance.
Therefore, Q will have moved a distance of 2π units after 1 second.
To determine the distance that Q will have moved 1 second after start-up, we need to calculate the length of the arc that Q traces as P moves around the unit circle counterclockwise.
1. Start by finding the circumference of the unit circle. The circumference of a circle is given by the formula C = 2πr, where r is the radius. In this case, since the unit circle has a radius of 1, the circumference is C = 2π(1) = 2π.
2. Next, determine the fraction of the circle completed in 1 second. If P moves at a constant rate counterclockwise around the circle, it completes one full revolution, which is equivalent to the circle's circumference, in 1 second. Therefore, the fraction of the circle completed in 1 second is 1/1 = 1.
3. Finally, calculate the distance that Q moves by multiplying the fraction of the circle completed (1) by the circumference of the circle (2π):
Distance moved by Q = 1 * 2π = 2π units.
Therefore, Q will have moved approximately 2π units, which is approximately 6.28 units (rounded to the nearest hundredth of a unit), after 1 second.