To solve the problem regarding the simple pendulum, we need to analyze the given angular displacement equation and extract the necessary information to find both the length of the pendulum and the velocity of the bob at a specific time. Let's break this down step by step.
Understanding Angular Displacement
The angular displacement of the pendulum is given by the equation:
θ = 0.1π sin(2πt) + π rad
Here, θ represents the angular displacement in radians, and t is the time in seconds. The term 0.1π sin(2πt) indicates that the pendulum undergoes simple harmonic motion (SHM) with a specific amplitude and frequency.
Finding the Length of the Pendulum
For a simple pendulum, the angular frequency (ω) is related to the length (L) of the pendulum by the formula:
ω = √(g/L)
From the angular displacement equation, we can identify the angular frequency:
ω = 2π rad/s
Now, we can set up the equation:
2π = √(g/L)
Where g is the acceleration due to gravity, approximately 9.81 m/s². Squaring both sides gives:
(2π)² = g/L
Rearranging for L, we have:
L = g / (2π)²
Substituting the value of g:
L = 9.81 / (2π)²
Calculating this gives:
L ≈ 0.25 m
Calculating the Velocity of the Bob
The velocity of the bob can be found by differentiating the angular displacement with respect to time:
v = dθ/dt
Taking the derivative of the angular displacement:
v = d/dt [0.1π sin(2πt) + π]
Using the chain rule, we get:
v = 0.1π * 2π * cos(2πt)
This simplifies to:
v = 0.2π² cos(2πt)
Now, we can find the velocity at t = 0.25 seconds:
v(0.25) = 0.2π² cos(2π * 0.25)
Since cos(π/2) = 0, we find:
v(0.25) = 0.2π² * 0 = 0 m/s
Summary of Results
- Length of the pendulum (L): 0.25 m
- Velocity of the bob at t = 0.25 sec: 0 m/s
In conclusion, the length of the simple pendulum is approximately 0.25 meters, and the velocity of the bob at 0.25 seconds is 0 m/s, indicating that the bob is at its maximum displacement at that moment.
