A pendulum consists of a small object hanging from the ceiling at the end of a string of negligible mass. The string has a length of 0.77 m. With the string hanging vertically, the object is given an initial velocity of 1.8 m/s parallel to the ground and swings upward in a circular arc. Eventually, the object comes to a momentary halt at a point where the string makes an angle θ with its initial vertical orientation and then swings back downward. Find the angle θ.

Question

A pendulum consists of a small object hanging from the ceiling at the end of a string of negligible mass. The string has a length of 0.77 m. With the string hanging vertically, the object is given an initial velocity of 1.8 m/s parallel to the ground and swings upward in a circular arc. Eventually, the object comes to a momentary halt at a point where the string makes an angle θ with its initial vertical orientation and then swings back downward. Find the angle θ.

Answer 1

To find the angle θ, we can use the law of conservation of mechanical energy. At the highest point of the pendulum swing, the object momentarily comes to a halt, which means all of its initial kinetic energy is converted to potential energy. At this point, the object is at its maximum height and the string makes an angle θ with its initial vertical orientation.

Let's use the law of conservation of mechanical energy:

Potential energy at highest point (PE) = Kinetic energy at initial point (KE)

The potential energy at the highest point is given by the formula: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.

Since the object is momentarily at rest, its kinetic energy at the initial point is 0.

Therefore, mgh = 0

Now, let's analyze the situation a bit further. At the highest point, the vertical component of the object's velocity is 0 since it momentarily comes to a halt. The only component of velocity remaining is the horizontal component, which is the initial velocity given as 1.8 m/s.

So, at the highest point, the string is acting as a hypotenuse of a right triangle. The vertical component of the string's length is the height (h), and the horizontal component is the distance traveled (d) parallel to the ground. We can use trigonometry to relate these components:

sin(θ) = h / d

We can rearrange this equation to solve for h:

h = d * sin(θ)

Now, we can substitute this value for h in the equation we obtained earlier:

mg * d * sin(θ) = 0

Now, we have two values, m and g, that we don't have in the problem statement. However, we can cancel them out as they exist on both sides of the equation. So, our equation becomes:

d * sin(θ) = 0

Since sin(θ) cannot equal 0 (as the string is not vertical anymore), we can divide both sides by sin(θ):

d = 0

Thus, the angle θ cannot be determined with the provided information because the equation d = 0 indicates that the string length becomes 0, which is not possible in this scenario.

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