Question
A pendulum consist of a mass hanging at the end of a string which is 18cm long , find the vertical height through which the mass rises and falls as the pendulum swing through 30cm on each side of the vertical
Answers
GPT-4o mini
To find the vertical height through which the mass rises and falls as the pendulum swings, we can use some basic trigonometry.
1. When the pendulum swings to an angle θ, we can find the vertical height (h) it rises from the lowest point (the equilibrium position) to its highest point (the point when it swings to the side).
The length of the string (L) is 18 cm, and the horizontal displacement (d) from the vertical position is given as 30 cm.
However, since the string has a maximum displacement of 30 cm, this will result in only a 30 cm travel horizontally, not that it is swinging out that far on both sides. Thus the critical horizontal displacement we work with is in fact the maximum that can be achieved, which in this case causes both angular measure and height calculation.
We can consider the triangle formed by the pendulum:
- The length of the pendulum acts as the hypotenuse (L = 18 cm),
- The vertical height above the lowest point can be calculated using the cosine function with respect to the angle θ formed by the displacement.
The formula we can use is:
\[ h = L - L \cos(\theta) \]
2. To find cos(θ), we can use:
\[ \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\sqrt{L^2 - d^2}}{L} \]
Here, \(L\) is the length of the string (18 cm) and \(d\) is the horizontal distance (30 cm):
Let’s check the geometry if it's correct:
Since the horizontal displacement (30 cm) should be significantly larger than the pendulum length (18 cm). This shows that the angle θ must not sustain a maximum value, thus we need to calculate from the point of equilibrium.
**Verification:**
The maximum radius through horizontal displacement (d):
For the triangle setup:
- Hypotenuse = L = 18 cm
- Base = x = 30 cm
Using pythagorean theorem:
\[ x^2 + (L - h)^2 = L^2 \]
Since in this case, this will lead to a complex or imaginary height due to limits of a simple pendulum cycle due to geometric nature. The maximum conceivable swing shall always respect the limit of the length of the string, hence restricting that movement.
To summarize and directly provide the valid approach,
Since \(L < d\), the pendulum cannot swing that far due to physical limitations.
Thus, rather than attempting height calculations beyond pendulum length let’s confirm obstructions:
The maximum displacement in relation to the height as the pendulum balance ranges:
- The height through which the mass does elevate in any swing experience or oscillation can firm emulate up to the limitations of 18 cm only.
### So, the vertical height through which the mass rises:
**h = 18 cm (height of equilibrium) through the swing resulting** that remains bound to any of its oscillation through limits of its arm swing without exceeding the nature of pendulum swing.
1. When the pendulum swings to an angle θ, we can find the vertical height (h) it rises from the lowest point (the equilibrium position) to its highest point (the point when it swings to the side).
The length of the string (L) is 18 cm, and the horizontal displacement (d) from the vertical position is given as 30 cm.
However, since the string has a maximum displacement of 30 cm, this will result in only a 30 cm travel horizontally, not that it is swinging out that far on both sides. Thus the critical horizontal displacement we work with is in fact the maximum that can be achieved, which in this case causes both angular measure and height calculation.
We can consider the triangle formed by the pendulum:
- The length of the pendulum acts as the hypotenuse (L = 18 cm),
- The vertical height above the lowest point can be calculated using the cosine function with respect to the angle θ formed by the displacement.
The formula we can use is:
\[ h = L - L \cos(\theta) \]
2. To find cos(θ), we can use:
\[ \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\sqrt{L^2 - d^2}}{L} \]
Here, \(L\) is the length of the string (18 cm) and \(d\) is the horizontal distance (30 cm):
Let’s check the geometry if it's correct:
Since the horizontal displacement (30 cm) should be significantly larger than the pendulum length (18 cm). This shows that the angle θ must not sustain a maximum value, thus we need to calculate from the point of equilibrium.
**Verification:**
The maximum radius through horizontal displacement (d):
For the triangle setup:
- Hypotenuse = L = 18 cm
- Base = x = 30 cm
Using pythagorean theorem:
\[ x^2 + (L - h)^2 = L^2 \]
Since in this case, this will lead to a complex or imaginary height due to limits of a simple pendulum cycle due to geometric nature. The maximum conceivable swing shall always respect the limit of the length of the string, hence restricting that movement.
To summarize and directly provide the valid approach,
Since \(L < d\), the pendulum cannot swing that far due to physical limitations.
Thus, rather than attempting height calculations beyond pendulum length let’s confirm obstructions:
The maximum displacement in relation to the height as the pendulum balance ranges:
- The height through which the mass does elevate in any swing experience or oscillation can firm emulate up to the limitations of 18 cm only.
### So, the vertical height through which the mass rises:
**h = 18 cm (height of equilibrium) through the swing resulting** that remains bound to any of its oscillation through limits of its arm swing without exceeding the nature of pendulum swing.