Tension problem - bead sliding on a string attached to a pole

[chelsea526]

Homework Statement

A 100 g bead is free to slide along an 80 cm long piece of string ABC. The ends of the string are attached to a vertical pole at A and C, which are 40 cm apart. When the pole is rotated about its axis, AB becomes horizontal.

Homework Equations

F = ma
Trigonometry formuals

The Attempt at a Solution

I am having trigonometry troubles! I only know one angle (90), and one side length (0.40 m). The bead is somewhere along the 0.80 m string, which is where the unknown angle is formed.

I thought I could use cos90 = 0.40/hyp, however, my calculator doesn't like this formula.
How can I find the lengths of the sides, so that I may then find the unknown angle??

 

[Doc Al]

I only know one angle (90), and one side length (0.40 m). The bead is somewhere along the 0.80 m string, which is where the unknown angle is formed.

Take advantage of the fact that you know the length of the string. What does that tell you?

 

[chelsea526]

Im not sure if I should be trying to use this equation

adj^2 + opp^2 = hyp^2 ?

I really am stumped that you don't know where along the string the bead is.

 

[Doc Al]

Im not sure if I should be trying to use this equation

adj^2 + opp^2 = hyp^2 ?

That's Pythagorean theorem. It'll do fine.

I really am stumped that you don't know where along the string the bead is.

Call one of the unknown sides X. What's the hypotenuse in terms of X?

 

[rwisz]

As a scientist, it is important to approach problems with a systematic and logical mindset. In this case, we can use the given information and equations to solve for the unknown angle and the tension on the string.

First, we can draw a diagram of the situation to visualize the problem. From the diagram, we can see that the string forms a right triangle with the pole. The unknown angle is the angle formed between the string and the horizontal line.

Next, we can use the trigonometric formula of cosθ = adjacent/hypotenuse to solve for the unknown angle. In this case, the adjacent side is the distance between the pole and the bead (40 cm), and the hypotenuse is the length of the string (80 cm). Plugging these values into the formula, we get cosθ = 40/80 = 0.5. Using a calculator, we can find the inverse cosine (cos^-1) of 0.5 to get the unknown angle, which is 60 degrees.

Now that we have the unknown angle, we can use the formula F = ma to solve for the tension on the string. The mass of the bead is given as 100 g, which is equal to 0.1 kg. We can assume that the bead is moving at a constant speed, so the acceleration (a) is equal to 0. Using the formula, we get Tension = 0.1 x 0 = 0 N.

Therefore, in this scenario, the tension on the string is 0 N and the unknown angle is 60 degrees. We can also use these values to calculate the force acting on the bead and the pole, but that is beyond the scope of this response.