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
(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.
To solve this problem, we need to understand the relationship between linear and angular speed and use the given information to calculate the required values. Let's go through each part step-by-step:
Part a) Finding dx/dt as a function of ϴ:
The linear speed of a point on the rim of a wheel can be calculated using the formula:
v = r * ω,
where v is the linear speed, r is the radius of the wheel, and ω is the angular speed in radians per unit time.
In this case, the radius (r) is given as 30 centimeters and the angular speed is 10 revolutions per second. To convert revolutions to radians, we use the fact that there are 2π radians in one revolution.
So, ω = (10 revolutions/second) * (2π radians/1 revolution) = 20π radians/second.
Now, we can substitute the values into the linear speed formula:
v = (30 cm) * (20π radians/second) = 600π cm/second.
The rate of change of x, dx/dt, is equal to the linear speed because x is the distance traveled by the point on the rim.
Therefore, dx/dt = 600π cm/second.
Part b) Graphing the function dx/dt as a function of ϴ:
To graph the function dx/dt as a function of ϴ, we'll plot points for different values of ϴ and connect them:
Let's choose some values for ϴ and calculate the corresponding dx/dt:
For ϴ = 0°, dx/dt = 0 cm/second.
For ϴ = 90°, dx/dt = 600π cm/second.
For ϴ = 180°, dx/dt = 0 cm/second.
For ϴ = 270°, dx/dt = -600π cm/second.
For ϴ = 360°, dx/dt = 0 cm/second.
These points form the shape of a sine wave when plotted on a graph. The graph can be generated using a graphing utility or software.
Part c) Finding when the absolute value of the rate of change of x is greatest and least:
From the graph, it can be observed that the absolute value of dx/dt is greatest when the sine wave reaches its maximum or minimum points. In this case, it occurs at ϴ = 90° and ϴ = 270°.
The absolute value of dx/dt is least when the sine wave crosses the x-axis, which happens at ϴ = 0° and ϴ = 180°.
Part d) Finding dx/dt when ϴ = 30° and ϴ = 60°:
To find dx/dt for specific values of ϴ, we use the equation we derived earlier:
dx/dt = 600π cm/second.
Therefore, when ϴ = 30° and ϴ = 60°, dx/dt remains constant at 600π cm/second.
Note: π represents the mathematical constant pi (approximately 3.14).
I hope this step-by-step explanation helps! Let me know if you have any further questions.
a) In order to find dx/dt as a function of θ, we need to relate the linear speed (dx/dt) to the angular speed (dθ/dt) using the formula:
Linear speed = Angular speed × Radius
Given that the wheel has a radius of 30 centimeters and is revolving at a rate of 10 revolutions per second, we can determine the angular speed:
Angular speed (dθ/dt) = 2π × Revolutions per second
Substituting the given values:
Angular speed (dθ/dt) = 2π × 10 revolutions per second
= 20π radians per second
Now, we can use the formula for linear speed:
dx/dt = (dθ/dt) × Radius
Substituting the values:
dx/dt = 20π radians per second × 30 centimeters
= 600π centimeters per second
So, dx/dt as a function of θ is given by:
dx/dt = 600π centimeters per second
b) Using a graphing utility, we can graph the function dx/dt = 600π centimeters per second. The x-axis represents θ, and the y-axis represents dx/dt in centimeters per second.
c) The absolute value of the rate of change of x is greatest when dx/dt is at its maximum value. In this case, the maximum value of dx/dt is 600π centimeters per second, which is constant for all values of θ. Therefore, the absolute value of the rate of change of x is greatest at all values of θ.
d) To find dx/dt when θ = 30° and θ = 60°, we can substitute the respective values into the equation:
When θ = 30°:
dx/dt = 600π centimeters per second
When θ = 60°:
dx/dt = 600π centimeters per second
So, at both θ = 30° and θ = 60°, dx/dt is equal to 600π centimeters per second.