(a) Let x be the x-coordinate of the point P with respect to the center of the wheel, such that x = r * cos(ϴ), where r is the radius (30 cm) and ϴ is the angle in radians. The angular speed, ω, is the rate of change of the angle over time, and we are given that the wheel revolves at a rate of 10 revolutions per second. To convert this rate to radians per second:
ω = 10 revolutions/second * 2π radians/revolution = 20π radians/second.
To find dx/dt, we need to find the derivative of x with respect to time. Using the chain rule, we have:
dx/dt = d(x)/d(ϴ) * d(ϴ)/dt = (-r * sin(ϴ)) * ω = -rω * sin(ϴ).
Since the radius, r, is 30 cm and the angular speed, ω, is 20π radians/second, our final expression for dx/dt is:
dx/dt = -30 * 20π * sin(ϴ) = -600π * sin(ϴ).
(b) In order to graph this function using a graphing utility, you can use the following steps:
1. Set your calculator or graphing program to radians mode.
2. input the function -600π * sin(ϴ) as Y1.
3. Set an appropriate window for your graph: one option is to choose a domain of 0 to 2π (since the angle ϴ is periodic with a period of 2π), and the range between -600π and 600π.
4. Graph the function, and observe the shape of the graph.
(c) From the graph of the function, we can see that the absolute value of the rate of change of x (i.e. the absolute value of dx/dt) is greatest when sin(ϴ) is equal to 1 or -1, and it is least when sin(ϴ) is equal to 0. This occurs when ϴ = π/2 and 3π/2 (greatest), and when ϴ = 0 and π (least).
(d) To find the values of dx/dt when ϴ = 30° and ϴ = 60°, we first convert these angles to radians:
30° = π/6 radians
60° = π/3 radians
Now, we plug in these values into our expression for dx/dt:
For ϴ = π/6:
dx/dt = -600π * sin(π/6) = -600π * 1/2 = -300π cm/s
For ϴ = π/3:
dx/dt = -600π * sin(π/3) = -600π * (√3/2) = -300π√3 cm/s
Hence, at ϴ = 30° (or π/6 radians), dx/dt is equal to -300π cm/s, and at ϴ = 60° (or π/3 radians), dx/dt is equal to -300π√3 cm/s.
LINEAR VS ANGULAR SPEED
A wheel of a radius 30 centimeters revolves at a rate of 10 revolutions per second.A dot is painted at a point P on the rim of the wheel.
a)find dx/dt as a function of ϴ.
b)use a graphing utility to graph a function in part (a)
c)when is the absolute value of the rate of change of x greatest?when is it least?
d)find dx/dt when ϴ=30° and ϴ=60°
show all your solutions pls thanks
1 answer