A variant on the carousel at a theme park is the swing ride. Swings are suspeneded from a rotating platform and move outward to form an angle x with the vertical as the ride rotates. The angle is related to the radial distance,r, in metres, from the centre of rotation; the acceleration, g=9.8 m/s^2 due to gravity; the speed,v, in metres per second, of the swing, accordon to the formula cotx=rg/v^2.
a) Determine the angle x for a swing located 3.5 m from the centre of rotation and moving at 5.4 m/s, to the nearest hundredths of a radian.
well, just plug in the numbers:
cot x = (3.5)(9.8)/5.4^2
To determine the angle x for a swing located 3.5 m from the centre of rotation and moving at 5.4 m/s, we will use the given formula
cot(x) = rg/v^2
First, let's substitute the values we are given:
r = 3.5 m
g = 9.8 m/s^2
v = 5.4 m/s
Plugging in these values into the formula:
cot(x) = (3.5 * 9.8) / (5.4)^2
cot(x) = 34.3 / 29.16
To find the value of x, we need to take the arccotangent of both sides of the equation. This can be written as:
x = arccot(34.3 / 29.16)
Using a calculator, we find
x ≈ 0.966 radians
Therefore, the angle x for a swing located 3.5 m from the centre of rotation and moving at 5.4 m/s is approximately 0.966 radians (rounded to the nearest hundredths).
To find the angle x, we can rearrange the given formula cotx = rg/v^2 and solve for x.
First, substitute the given values: r = 3.5 m and v = 5.4 m/s.
The formula becomes cotx = (3.5 * g) / (5.4^2).
Next, simplify the equation:
cotx = (3.5 * 9.8) / (5.4^2)
cotx = 34.3 / 29.16.
Now, take the inverse cotangent (also known as arccot) of both sides to isolate x:
x = arccot(34.3 / 29.16).
Use a scientific calculator or an online calculator to find the arccot of 34.3 / 29.16. Round the answer to the nearest hundredth of a radian.
Therefore, x = 0.91 radians (rounded to the nearest hundredth).
Therefore, the angle x for a swing located 3.5 m from the center of rotation and moving at 5.4 m/s is approximately 0.91 radians.