A carnival seat and a passenger has a combined weight of 193. As the seat is spun on a 18m rope tethered to a pole at an angle of 59.2 degrees, at what speed is the seat traveling?
I do not know where your angle is
If from straight down then:
T = tension
m g = weight
A = angle from vertical
vertical equation
T cos A = m g = m(9.81)
so
T = m(9.81)/cos A
T sin A = m Ac = m v^2/R
where R = 18 sin A
so
m (9.81) sin A = m v^2 /(18 sin A)
v^2 = 9.81 * 18 * sin^2 A
note, mass cancels, does not matter, otherwise carnival ride would not work :)
T = m(9.81)/cos A
T sin A = m Ac = m v^2/R
where R = 18 sin A
so
m (9.81) sin A/cosA = m v^2 /(18 sin A)
v^2 = 9.81 * 18 * sin^2 A/cos A
check my arithmetic !!!
The rope forms a triangle with the pole and the angle is between the rope and the point at which it connects to the pole.
ok, that is may angle A so plug sin 59.2 and cos 59.2 in
but check my algebra first
To find the speed at which the carnival seat is traveling, we can use the concept of centripetal force. Centripetal force is the force that keeps an object moving in a circular path. It depends on the mass of the object, the radius of the circular path, and the speed of the object.
In this case, we are given the combined weight of the seat and the passenger, which we can consider as the mass (m) of the object. The weight (W) is the force acting vertically downward due to gravity.
We also know that the seat is traveling in a circular path with a radius (r) equal to the length of the rope, which is 18m.
Now, let's break down the weight into its component forces. The weight force can be resolved into two forces: one perpendicular to the circular path (W⊥) and one tangential to the circular path (W//).
The force acting perpendicular to the circular path (W⊥) is equal to W * cosθ, where θ is the angle between the rope and the vertical line.
W⊥ = W * cosθ
The centripetal force (Fc) required to keep the seat moving in a circle is provided by the force acting perpendicular to the circular path (W⊥).
Fc = W⊥
Since we are given the combined weight (W), we can substitute the value of W⊥ to find the centripetal force:
Fc = W * cosθ
Now, we know that centripetal force (Fc) is given by the equation:
Fc = (m * v^2) / r
Where v is the speed at which the seat is traveling.
Combining the two equations, we get:
W * cosθ = (m * v^2) / r
Rearranging this equation, we can solve for v:
v = √(r * (W * cosθ) / m)
Plugging in the given values:
m = 193 (combined weight of the seat and the passenger)
r = 18m (length of the rope)
θ = 59.2 degrees (angle between the rope and the vertical line)
W = m * g (weight = mass * acceleration due to gravity, g ≈ 9.8 m/s^2)
Substituting these values into the equation, we can calculate the speed (v) at which the seat is traveling.