a horizontal position of the pendulum of a grandfather clock can be modelled by h(t)=Acos(2(pi)t/T), where A is amplitude of the pendulum in meters, t is time in seconds, and T is the period of the pendulum, in seconds. If the pendulum has a velocity of v(t)=sin Pi(t) find:
A) an equation for h(t)
B) a formula for acceleration of the pendulum at time t.
C) the maximum horizontal displacement
D) the time this displacement occurs.
E) the length of the pendulum, rounded to the nearest meter.
A) Equation for h(t):
The equation for h(t) is given as h(t) = Acos(2πt/T).
B) Formula for acceleration of the pendulum at time t:
Acceleration (a) in terms of velocity (v) is given as the first derivative of velocity with respect to time.
Since v(t) = sin(πt), we can differentiate v(t) with respect to t to find the acceleration.
a(t) = d/dt[v(t)] = d/dt[sin(πt)] = π*cos(πt)
Therefore, the formula for acceleration of the pendulum at time t is a(t) = π*cos(πt).
C) Maximum horizontal displacement:
Maximum horizontal displacement occurs when the value of |h(t)| is maximum.
Given that h(t) = Acos(2πt/T), the maximum value of |h(t)| is equal to the amplitude A.
Therefore, the maximum horizontal displacement is A.
D) Time at which maximum displacement occurs:
We can find the time at which the maximum displacement occurs by finding the value of t that maximizes h(t) = Acos(2πt/T).
To maximize h(t), we want to find the value of t for which cos(2πt/T) = 1 (maximum value of cosine).
Since cos(2πt/T) = 1, we have 2πt/T = 0 (cos(0) = 1).
Solving for t, we get t = 0.
Therefore, the time at which the maximum displacement occurs is t = 0.
E) Length of the pendulum:
The period of the pendulum (T) is the time it takes for the pendulum to complete one full cycle.
Since the pendulum completes one full cycle when cos(2πt/T) = 1, we can set 2πt/T = 2π (one full cycle).
Simplifying the equation, we have t = T.
Therefore, the length of the pendulum (L) can be calculated using the formula:
L = gT^2/(4π^2)
Here, g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the values, and rounding to the nearest meter, we can find the length of the pendulum.
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(2πt/T)
To find equation for h(t), we need to integrate the velocity equation, v(t) = sin(πt), with respect to time.
∫v(t) dt = ∫sin(πt) dt
Applying the integral, we get:
h(t) = -cos(πt) + 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(π(0)) + C
A = -1 + C
C = A + 1
Therefore, the equation for h(t) is:
h(t) = -cos(πt) + (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(πt))/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(πt) + (A + 1)
At t = 0, h(0) = -cos(π(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/(2π))^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/(2π))^2 * 9.8
L = (1/(π))^2 * 9.8
L ≈ 9.9 meters (rounded to the nearest meter).
Therefore, the length of the pendulum is approximately 10 meters.
v = dh/dt = sin pi t
then h = - (1/pi)cos pi t
but h = A cos (2 pi t/T)
so
A = -1/pi
pi = 2 pi /T so T = 2
so in the ean
h =-(1/pi) cos pit
= -(1/pi)cos(2 pi t/2)
B)
h =-(1/pi) cos pit
v = dh/dt = sin pi t
a = dv/dt = pi cos pi t
C)1/pi
D) cos 0 = 1 so at t = 0 , 1 , 2 etc
E) T = 2 pi sqrt(L/g)
here T = 2
2 = 2 pi sqrt (L/g)
sqrt (L/g) = 1/pi
L/g = 1/pi^2)
L = g/pi^2