Forces and centripital acceleration

[benji]

A "swing" ride at a carnival consists of chairs that are swung in a circle by 12.0-m cables attached to a vertical pole; the pole and the cable make an angle of 65.0 degrees above the horizontal. Suppose the total mass of a chair and its occupant is 220 kg. (a) Determine the tension in the cable attached to the chair. (b) Find the speed of the chair.

I can't figure it out!

 

[Vector Sum]

THis is a great problem, but maybe not for the students!

Centripetal force must be provided by some other recognizable force. In this case, it is provided by the horizontal componant of the tension.

The vertical componant of the tension has to balance the only other force on the chair (what could it be?).

You have the angle of the cable, and you can calculate the weight of the chair. Can you take it from there?

 

[wais]

Don't worry, understanding forces and centripetal acceleration can be tricky at first. Let's break down the problem and go through the steps to solve it.

First, we need to understand the forces acting on the chair and its occupant. In this case, the only force acting on them is the tension force in the cable. This is because the chair is moving in a circular motion, so there must be a force pulling it towards the center of the circle to keep it moving.

(a) To determine the tension in the cable, we can use Newton's second law, which states that the net force on an object is equal to its mass times its acceleration (F=ma). In this case, the net force is the tension in the cable, the mass is 220 kg, and the acceleration is the centripetal acceleration, which is given by the formula a = v^2/r, where v is the speed of the chair and r is the radius of the circle (in this case, the length of the cable).

So, we can rewrite the equation as T = m(v^2/r), where T is the tension in the cable. Now, we just need to plug in the values given in the problem. The radius of the circle is the length of the cable, which is 12.0 m. The mass is 220 kg. We are looking for the speed, so we will leave that as v. The equation now looks like this:

T = (220 kg)(v^2/12.0 m)

To solve for T, we need to know the value of v. So, let's move on to part (b) to find that.

(b) To find the speed of the chair, we can use the fact that the chair is moving in a circular motion. The formula for the speed of an object moving in a circle is v = ωr, where ω is the angular velocity (which is equal to v/r) and r is the radius of the circle. In this case, we know the angle between the cable and the horizontal is 65.0 degrees, so the angle between the cable and the vertical is 25.0 degrees (since the two angles must add up to 90 degrees). This means that ω = v/12.0 m = tan(25.0 degrees). Solving for v, we get v = 12.0 m tan(25.0 degrees