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. (è = 60.0°) Suppose the total mass of a chair and its occupant is 222 kg.
To find the tension in the cable, you can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
In this case, the net force acting on the chair is the centripetal force, which is provided by the tension in the cable.
The formula for centripetal force is:
F = m * a
where F is the centripetal force, m is the mass of the object, and a is the acceleration.
Since the chair is moving in a circle, its acceleration is given by the formula:
a = ω² * r
where a is the acceleration, ω (omega) is the angular velocity, and r is the radius of the circular path.
The angular velocity ω is the rate of change of angle with respect to time and is related to the period T (time taken to complete one full revolution) by the formula:
ω = 2π / T
where ω is the angular velocity and T is the period.
In this case, the angle given is 60.0°, which is one-sixth of a full revolution. Therefore, the period T is equal to 1/6 of the time taken for one full revolution.
To find the period T, you need to know the time taken for one full revolution. Let's assume it takes 10 seconds for one full revolution.
So, T = 1/6 * 10 = 1.67 seconds
Using this value for T, you can calculate the angular velocity ω:
ω = 2π / 1.67 = 3.77 radians/second
Now you have the value of ω and the radius of the circular path, which is given as 12.0 m.
Using these values, you can calculate the acceleration a:
a = (3.77)² * 12.0 = 180.63 m/s²
Finally, you can substitute the mass of the chair and occupant (m = 222 kg) and the calculated acceleration (a = 180.63 m/s²) into the formula for centripetal force:
F = m * a = 222 * 180.63 = 40,135.86 N
Therefore, the tension in the cable is approximately 40,136 N.