To answer these questions, we need to derive the required equations from the given information. Let's go step by step:
A) Equation for h(t):
Given: h(t) = Acos(2Ï€t/T)
To find equation for h(t), we need to integrate the velocity equation, v(t) = sin(Ï€t), with respect to time.
∫v(t) dt = ∫sin(πt) dt
Applying the integral, we get:
h(t) = -cos(Ï€t) + C
To find the value of C, we can use the initial condition that the pendulum starts at its maximum displacement, h(0) = A.
h(0) = -cos(Ï€(0)) + C
A = -1 + C
C = A + 1
Therefore, the equation for h(t) is:
h(t) = -cos(Ï€t) + (A + 1)
B) Formula for acceleration, a(t):
Since acceleration is the derivative of velocity, we need to differentiate the velocity equation with respect to time.
a(t) = d(v(t))/dt
a(t) = d(sin(Ï€t))/dt
Differentiating, we get:
a(t) = πcos(πt)
Therefore, the formula for acceleration of the pendulum at time t is:
a(t) = πcos(πt)
C) Maximum horizontal displacement:
To find the maximum horizontal displacement, we need to find the maximum value of h(t) in one period, which occurs at t = 0.
h(t) = -cos(Ï€t) + (A + 1)
At t = 0, h(0) = -cos(Ï€(0)) + (A + 1)
h(0) = -(1) + (A + 1)
h(0) = A
Therefore, the maximum horizontal displacement is A.
D) Time of maximum displacement:
We already have the time of maximum displacement, which is t = 0.
E) Length of the pendulum:
The period of a pendulum, T, is the time taken for one complete oscillation. In this case, it is given by T = 2 seconds.
The formula for the period, T, is related to the length of the pendulum, L, through the equation:
T = 2π√(L/g)
Solving for L:
L = (T/(2Ï€))^2 * g
Given T = 2 seconds and taking average value of acceleration due to gravity g = 9.8 m/s^2, we can calculate the length of the pendulum:
L = (2/(2Ï€))^2 * 9.8
L = (1/(Ï€))^2 * 9.8
L ≈ 9.9 meters (rounded to the nearest meter).
Therefore, the length of the pendulum is approximately 10 meters.