A 2kg mass is attached to a 5.50m long string. The mass is pulled 30 degrees from its lowest point and then released. What is the initial height of the pendulum above its lowest point?
I think I should be solving for Lcos(theta), where L = 5.50 and theta = 30, but that gives me the wrong answer.
Why? What is the correct procedure, and how might I visualize it in terms of right triangles?
what is 5.5(1-cos30) ? Draw the figure, goodness. you have a Lcos(theta) along the vertical, but that is not all the way down.
Thank you! I drew the figure and see the differences in the length, but still why is it 1 - cos30?
ok, goodness. it is 5.5-5.5cos30.
oh wow, thanks. I see that.
To solve this problem, you can use trigonometry and the concept of right triangles. Let's break it down step by step:
1. Start by visualizing the problem. Imagine a pendulum consisting of a mass attached to a string. The length of the string is given as 5.50m.
2. The pendulum is pulled away from its lowest point at an angle of 30 degrees relative to the vertical line. This means that the angle between the string and the vertical line is 30 degrees.
3. We need to find the initial height of the pendulum above its lowest point. To do this, we can use the concept of right triangles and trigonometry.
4. Draw a right triangle with the vertical line being the hypotenuse. Label one side of the triangle as h (the height we need to find) and the other side as L (the length of the string).
5. The angle between the vertical line and the hypotenuse is 30 degrees, so label this angle as theta.
6. To find the initial height (h), we need to consider the horizontal component of the string. In this case, it is given by L * cos(theta).
7. Plug in the values: L = 5.50m and theta = 30 degrees. Calculate L * cos(theta) to find the initial height above the lowest point.
Now, let's evaluate your initial attempt of solving the problem using L * cos(theta) and see why it may have given you the wrong answer.
If we use L * cos(theta), we get:
h = L * cos(theta) = 5.50 * cos(30°) ≈ 5.50 * 0.866 = 4.762m
However, this value is not the correct answer. The reason is that L * cos(theta) gives the horizontal component of the string, not the vertical height above the lowest point.
To find the initial height, we need to consider the vertical component of the string, which is given by L * sin(theta).
So, the correct procedure is to use L * sin(theta) instead of L * cos(theta).
Let's calculate the correct initial height using L * sin(theta):
h = L * sin(theta) = 5.50 * sin(30°) ≈ 5.50 * 0.5 = 2.75m
Therefore, the correct initial height of the pendulum above its lowest point is approximately 2.75 meters.