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2 answers:
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The equation of motion of a pendulum is:
where it its length and
is the gravitational acceleration. Notice that the mass is absent from the equation! This is quite hard to solve, but for <em>small</em> angles (
), we can use:
Additionally, let us define:
We can now write:
The solution to this differential equation is:
where and
are constants to be determined using the initial conditions. Notice that they will not have any influence on the period, since it is given simply by:
This justifies that the period depends only on the pendulum's length.
Answer:
a) Gravitational potential energy = 399 J
b) Gravitational potential energy = 66.5 J
c) Gravitational potential energy = 0 J
Explanation:
Hi there!
Please, see the attached figure for a better understanding of the problem.
a) When the ropes are horizontal, the height of the child, relative to the child's lowest position, is 2.10 m (see figure).
The gravitational potential energy is calculated as follows:
PE = mgh
Where:
PE = potential energy.
mg = weight of the child
h = height.
Then when the ropes are horizontal, the potential energy will be:
PE = 190 N · 2.10 m = 399 J
b) When the ropes make a 34.0° with the vertical, the height of the child is 2.10 m minus x (see figure). To find x, we can use trigonometry of right triangles:
cos angle = adjacent side / hypotenuse
cos 34.0° = x / 2.10 m
x = 2.10 m · cos 34.0° = 1.75 m
Then, the height of the child relative to the lowest position is
(2.10 m - 1.75 m) = 0.35 m
Therefore, the gravitational potential energy will be:
PE = 190 N · 0.35 m
PE = 66.5 J
c) When the child is at the bottom of the circular arc the height is zero (the child is at the lowest position), then, the gravitational potential energy will be zero.
