A "swing" ride at a carnival consists of chairs that are swung in a circle by 12.0 m cables attached to a vertical rotating pole, as the drawing shows. (θ = 56.0°) Suppose the total mass of a chair and its occupant is 190 kg.
(a) Determine the tension in the cable attached to the chair.
(b) Find the speed of the chair.
First, let's find the length of the vertical component of the cable. The length of the vertical component can be found by:
Length = 12 m * cos(56°)
Next, we'll find the horizontal component of the cable's tension. At the top of the swing, the centripetal force acting on the system is equal to the horizontal component of tension. So,
Fc = T * sin(θ)
mv^2 / r = T * sin(θ)
where m = 190 kg, r = 12 m, and θ = 56°
(a) To find the tension in the cable, first consider the vertical component of the force. The tension in the cable is equal to the gravitational force acting on the chair and its occupant:
T * cos(θ) = mg
T = mg / cos(θ)
T = 190 kg * 9.81 m/s^2 / cos(56°)
T = 1680 N
(b) To find the speed of the chair, we will use the centripetal force equation we found earlier:
mv^2 / r = T * sin(θ)
Solve for v,
v^2 = T * sin(θ) * r / m
v = sqrt((1680 N * sin(56°) * 12 m) / 190 kg)
v = 9.21 m/s
The tension in the cable is 1680 N, and the speed of the chair is 9.21 m/s.
To determine the tension in the cable attached to the chair, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the net force is provided by the tension in the cable and is directed towards the center of the circular path. The acceleration of the chair can be found using centripetal acceleration formula:
a = v^2 / r
Where:
a is the centripetal acceleration
v is the velocity of the chair
r is the radius of the circular path
The radius of the circular path can be found using trigonometry. In the given diagram, the radius is represented by the distance between the vertical pole and the cable. The angle between the cable and the vertical pole is given as θ = 56.0°.
Using trigonometry, we can find that the radius r is equal to the length of the cable multiplied by the sine of the angle:
r = 12.0 m * sin(56.0°)
Now we need to find the velocity of the chair. The velocity can be found using the formula:
v = ω * r
where:
v is the velocity
ω (omega) is the angular velocity
r is the radius
The angular velocity can be found using the formula:
ω = √(g / r)
where:
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Now we have all the necessary information to solve the problem. Substitute the values into the equations:
r = 12.0 m * sin(56.0°)
ω = √(9.8 m/s^2 / r)
v = ω * r
(a) To calculate the tension in the cable attached to the chair, we need to find the net force acting on the chair. The net force can be calculated using Newton's second law:
F = m * a
where:
F is the net force
m is the mass of the chair and occupant (190 kg)
a is the centripetal acceleration
(b) To calculate the speed of the chair, use the formula:
v = ω * r
where:
v is the speed of the chair
ω is the angular velocity
r is the radius of the circular path
Plug in the values and solve for the tension (a) and speed (b) accordingly.
To solve this problem, we'll break it down into two main steps: calculating the tension in the cable attached to the chair and finding the speed of the chair.
Step 1: Calculating the tension in the cable attached to the chair
We'll start by analyzing the forces acting on the chair at the highest point of the swing, where the tension in the cable is maximum and the acceleration due to gravity is pointing downwards.
At this point, the vertical component of the tension is equal to the weight of the system (chair + occupant):
T · cos(θ) = m · g
where:
T is the tension in the cable attached to the chair
θ is the angle between the cable and the vertical axis (given as 56.0°)
m is the mass of the chair and its occupant (given as 190 kg)
g is the acceleration due to gravity (approximately 9.8 m/s²)
Substituting the given values into the equation, we get:
T · cos(56.0°) = (190 kg) · (9.8 m/s²)
Now, let's solve for T by dividing both sides of the equation by cos(56.0°):
T = (190 kg) · (9.8 m/s²) / cos(56.0°)
Using a calculator, we find that T ≈ 2046 N (rounded to the nearest whole number).
Therefore, the tension in the cable attached to the chair is approximately 2046 N.
Step 2: Finding the speed of the chair
To find the speed of the chair, we can use the fact that the tension in the cable provides the centripetal force required to keep the chair moving in a circular path.
The centripetal force, provided by the tension in the cable, is given by:
F_c = m · v² / r
where:
F_c is the centripetal force
m is the mass of the chair and its occupant (given as 190 kg)
v is the speed of the chair (to be determined)
r is the radius of the circular path (given as 12.0 m)
Since the centripetal force is equal to the tension in the cable, we can write:
T = m · v² / r
Substituting the known values, we have:
2046 N = (190 kg) · v² / 12.0 m
Now, let's solve for v²:
v² = (2046 N) · (12.0 m) / (190 kg)
Using a calculator, we find that v² ≈ 129.16 m²/s² (rounded to two decimal places).
Taking the square root of both sides, we get:
v ≈ sqrt(129.16 m²/s²)
v ≈ 11.36 m/s (rounded to two decimal places)
Therefore, the speed of the chair is approximately 11.36 m/s.