A “swing” ride at a carnival consists of chairs that are swung in a circle by 13.5 m long cables attached to a vertical rotating pole, as the drawing shows. Note that the cables make an angle of 60.0o with respect to the vertical, as shown. Suppose the total mass of a chair and its occupant is 125 kg, treated as a point mass at the end of the cable.
(a) Draw a free-body diagram for the chair/occupant (treated as a point mass, m = 125 kg). Indicate the coordinate system you will use to solve the problem.
(b) Determine the magnitude of the tension in the cable attached to the chair. (c) Find the speed of the chair.
(a) Alright, let's get creative and draw a free-body diagram for our swinging amusement park enthusiast. We have our point mass, the chair and its occupant, let's call them Chuck, hanging from the 13.5 m long cable. Chuck is going for a joy ride with an angle of 60.0o from the vertical. To solve this problem, let's go with a coordinate system where the vertical axis is positive upwards and the horizontal axis is parallel to the ground.
(b) Now, let's calculate the magnitude of the tension in the cable. Since Chuck is swinging in a circle, we know that there must be a centripetal force acting towards the center of the circle. In this case, the tension in the cable provides that force. So, Tension is our hero in this story, keeping Chuck from flying off into the wild blue yonder!
(c) To find the speed of the chair, we can use the concept of centripetal force. The tension in the cable, which we just calculated, is the centripetal force. And we know that the centripetal force is given by the formula Fc = mass x acceleration, where the acceleration is v^2 / r. Here, v represents the speed of the chair and r is the radius of the circular path, which is the length of the cable.
But wait, we don't know the acceleration yet. Well, it just so happens that the acceleration is related to the speed by the equation a = v^2 / r. So, we can substitute that in the centripetal force equation. And now we have a new equation, Tension = mass x (v^2 / r). Now we can solve for the speed, v.
No worries, I'll do the calculation for you. Give me a moment...
(a) To draw a free-body diagram for the chair/occupant, we need to consider the forces acting on it. The only forces acting on the chair/occupant are the tension in the cable (T) and the force of gravity (mg).
In the free-body diagram, we can represent the chair/occupant as a single dot (point mass) with downward force mg acting at its center. The tension T in the cable will be directed at an angle of 60.0° with respect to the vertical.
The coordinate system we will use is the Cartesian coordinate system, with the x-axis pointing horizontally to the right and the y-axis pointing vertically upward.
(b) To determine the magnitude of the tension in the cable attached to the chair, we need to analyze the forces acting on the chair/occupant in the vertical direction. Using Newton's second law, we can set up the following equation:
∑Fy = T - mg = 0
We can rearrange the equation to solve for the tension T:
T = mg
Given that the mass of the chair/occupant is 125 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can substitute these values into the equation:
T = (125 kg)(9.8 m/s^2) = 1225 N
Therefore, the magnitude of the tension in the cable attached to the chair is 1225 N.
(c) To find the speed of the chair, we can analyze the forces acting on the chair/occupant in the horizontal direction. The only force in the horizontal direction is the tension in the cable (T).
Using the relationship between the net force and centripetal force:
F_c = m * a_c
where F_c is the centripetal force, m is the mass of the chair/occupant, and a_c is the centripetal acceleration.
The centripetal force is equal to the tension in the cable (T), and the centripetal acceleration can be calculated using the equation:
a_c = v^2 / r
Where v is the speed of the chair and r is the radius of the circular path (which is equal to the length of the cable, 13.5 m).
Substituting the values into the equation:
T = m * v^2 / r
Solving for v:
v^2 = (T * r) / m
v = sqrt((T * r) / m)
Substituting the given values:
v = sqrt((1225 N * 13.5 m) / 125 kg)
v = sqrt(16537.5 Nm / 125 kg)
v = sqrt(132.3 m^2/s^2)
v ≈ 11.5 m/s
Therefore, the speed of the chair is approximately 11.5 m/s.
To answer the given questions, we need to first understand the forces acting on the chair/occupant in the swing ride.
(a) Free-Body Diagram:
- Gravity (Weight): It acts vertically downwards with a magnitude equal to the mass of the chair/occupant (125 kg) multiplied by the acceleration due to gravity (9.8 m/s^2).
- Tension: It acts along the cable and pulls the chair/occupant towards the center of the circular path.
To solve the problem, we will use a coordinate system with the x-axis horizontal and pointing to the right, and the y-axis vertical and pointing upwards.
Now, let's move on to the next question.
(b) Magnitude of Tension in the Cable:
In order to determine the tension in the cable attached to the chair, we need to consider the forces acting on the chair in the vertical direction (along the y-axis).
The tension in the cable will be equal to the force required to balance the weight of the chair/occupant in the vertical direction.
Using trigonometry, we can calculate the vertical component of the tension. The vertical component of the tension can be found using the formula:
T * cosθ = m * g
Where:
T is the tension in the cable
θ is the angle the cable makes with respect to the vertical (60.0 degrees in this case)
m is the mass of the chair/occupant (125 kg)
g is the acceleration due to gravity (9.8 m/s^2)
Substituting the given values into the equation, we can solve for T.
T * cos(60.0 degrees) = (125 kg) * (9.8 m/s^2)
T * (0.5) = 1225 N
T = 1225 N / (0.5)
T = 2450 N
Therefore, the magnitude of the tension in the cable attached to the chair is 2450 Newtons (N).
Now, let's move on to the final question.
(c) Speed of the Chair:
To find the speed of the chair, we can use the concept of centripetal force.
The tension in the cable provides the centripetal force required to keep the chair moving in a circular path. The centripetal force is given by:
T = (m * v^2) / r
Where:
T is the tension in the cable (2450 N)
m is the mass of the chair/occupant (125 kg)
v is the speed of the chair (what we need to find)
r is the radius of the circular path (given as 13.5 m)
We can rearrange the equation to solve for v:
v^2 = (T * r) / m
v = sqrt((T * r) / m)
Substituting the given values into the equation, we can find the speed of the chair.
v = sqrt((2450 N * 13.5 m) / 125 kg)
v = sqrt(26460 m^2/s^2 / 125)
v = sqrt(211.68 m^2/s^2)
v = 14.54 m/s
Therefore, the speed of the chair is approximately 14.54 m/s.
I hope this explanation helps you understand how to solve the problem! Let me know if you have any further questions.