The “Giant Swing” ride consists of a vertical shaft with a number of horizontal arms attached at the top as shown. A rigid cable 5.0 m long connects a seat to the arm at a point 2.0 m from the center shaft. When the ride reaches its full speed, the
angle made by the arm is 35.0o.
(i) What is the time that it takes for one full revolution at the maximum speed?
(ii) If a 60.0-kg person goes on the ride, and we ignore the mass and weight of the metal shaft, then what is the tension in the 5.00 m cable?
(i) We can find the maximum speed that the swing rotates by finding the horizontal component of motion. The radius of the circle traced by the seat is:
r = 2.0 m * sin(35°) = 1.14 m
Now we can find the linear velocity (v) using centripetal acceleration:
a = v^2 / r
For an object in circular motion, the centripetal acceleration is:
a = 4 * pi^2 * r / T^2
where T is the time period of one full revolution. Substituting the value of r, we get:
T^2 = 4 * pi^2 * (1.14 m) / a
Now, the centripetal acceleration can also be expressed using the angle (theta) and the radius (R) of the arm:
a = R * g * sin(theta)
We know R = 2.0 m, g = 9.81 m/s^2, and theta = 35°:
a = 2.0 m * 9.81 m/s^2 * sin(35°) = 11.36 m/s^2
Substituting the value of a into the equation for T, we get:
T^2 = 4 * pi^2 * (1.14 m) / 11.36 m/s^2
T^2 ≈ 3.996 s^2
Taking the square root of both sides, we find:
T = 2.0 s
So it takes 2.0 seconds for one full revolution at the maximum speed.
(ii) We can find the tension in the cable by considering the forces acting on the person. There are two forces acting on the person: the gravitational force (Fg) acting downward and the tension (T) acting along the cable.
We can divide the tension into two components: one acting horizontally (Tx) and another acting vertically (Ty). The sum of the forces in the vertical direction should be equal to the centripetal force, and the Ty should balance the person's weight.
Ty = Fg = m * g = 60 kg * 9.81 m/s^2 = 588.6 N
The centripetal force is given by:
Fc = m * a = 60 kg * 11.36 m/s^2 = 681.6 N
Now, the horizontal component of the force can be expressed using the angle:
Tx = Fc / cos(theta) = 681.6 N / cos(35°) = 831.75 N
The tension in the cable is the vector sum of the horizontal and vertical components of the force:
T = sqrt(Tx^2 + Ty^2) = sqrt(831.75 N^2 + 588.6 N^2) ≈ 1030 N
So the tension in the cable is approximately 1030 N.
To solve this problem, we can use the concepts of circular motion and centripetal force.
(i) To find the time it takes for one full revolution at maximum speed, we can use the equation:
Period (T) = 2π / ω
where ω is the angular velocity.
The angular velocity can be found using the formula:
ω = θ / t
where θ is the angle traveled and t is the time taken.
Given that the angle made by the arm is 35.0°, we convert it to radians:
θ = 35.0° × (π / 180°) = 0.6109 radians
We can assume that the time taken for one full revolution is the same as the time taken for the arm to make an angle of 2π radians.
Therefore, θ = 2π.
Rearranging the formula for angular velocity, we get:
ω = θ / t
t = θ / ω = 2π / ω = 2π / (0.6109 radians) = 10.30 seconds
So, it takes approximately 10.30 seconds for one full revolution at maximum speed.
(ii) To calculate the tension in the cable, we need to consider the centripetal force acting on the person.
The centripetal force (Fc) can be calculated using the formula:
Fc = m * ω² * r
where m is the mass of the person, and r is the radius of the circular path (5.0 m - 2.0 m = 3.0 m).
Given that the mass of the person is 60.0 kg, we have:
Fc = 60.0 kg * (0.6109 radians / s)² * 3.0 m
Fc ≈ 66.36 N
Therefore, the tension in the 5.00 m cable is approximately 66.36 N.
To solve these problems, we will need to use some concepts from physics, specifically circular motion and force analysis. Let's break down each part of the question:
(i) Time for one full revolution at maximum speed:
To find the time for one full revolution, we need to know the angular velocity. The formula for angular velocity is ω = Δθ/Δt, where ω is the angular velocity, Δθ is the change in angle, and Δt is the time taken for that change. In this case, the maximum angle made by the arm is 35.0 degrees.
To find the angular velocity, we need to convert this angle to radians. 1 radian is equal to 180/π degrees. So, 35.0 degrees is equal to (35.0 × π)/180 radians.
Next, we need to determine the time taken for one full revolution, which is the time taken for the arm to move through 2π radians. We can use the equation ω = Δθ/Δt to rearrange and solve for Δt.
ω = (2π radians) / Δt
Rearranging, we get Δt = (2π) / ω
Substituting the value of ω we calculated earlier, we can find the time taken for one full revolution.
(ii) Tension in the cable:
To find the tension in the cable, we need to consider the forces acting on the person. The two main forces are the tension in the cable and the gravitational force acting downward.
Using Newton's second law, F = ma, where F is the net force, m is the mass, and a is the acceleration, we can find the net force acting on the person.
The net force acting on the person is equal to the tension in the cable minus the force due to gravity, which is given by F = mg, where g is the acceleration due to gravity.
Setting these forces equal to each other, we have:
Tension - mg = ma
Simplifying, we get:
Tension = m(g + a)
Now, the person is undergoing circular motion, so we need to find the centripetal acceleration, which is given by ac = v^2 / r, where v is the velocity and r is the radius of the motion.
Since we are given the length of the cable (5.0 m) and the distance from the center shaft to the attachment point (2.0 m), we can find the radius using the Pythagorean theorem:
r = √(5.0^2 - 2.0^2)
Next, we need to find the velocity of the person. The velocity of an object undergoing circular motion can be found using the formula v = ωr, where ω is the angular velocity and r is the radius.
Once we have the radius and velocity, we can calculate the centripetal acceleration using ac = v^2 / r.
Finally, we substitute the values of acceleration, mass, and acceleration due to gravity into the equation Tension = m(g + a) to find the tension in the cable.
By following these steps, we can calculate the values for both (i) and (ii) of the given question.