Two physical pendulums (not simple pendulums) are made from meter sticks that are suspended from the ceiling at one end. The sticks are uniform and are identical in all respects, except that one is made of wood (mass=0.34 kg) and the other of metal (mass=0.63 kg) They are set into oscillation and execute simple harmonic motion. Determine the period of (a) the wood pendulum and (b) the metal pendulum
We know that
omega = 2*pi*freq = sqrt(mgL/I), where m is the mass, g is the acceleration due to gravity, and I is the moment of inertia.
So, if we can evaluate the equation above, we can get the frequency, and then take 1/freq in order to get the period.
However, don't we need the moment of Intertia, I, for the meter stick?
How do we solve this?
Thanks!
To determine the period of the pendulums, we need to calculate the moment of inertia, I, for each pendulum.
The moment of inertia for a uniform rod rotating about one end (suspended from that end) can be calculated as:
I = (1/3) * m * L^2,
where m is the mass of the rod and L is the length of the rod.
(a) For the wood pendulum:
Given that the mass of the wood pendulum is 0.34 kg and the length of the meter stick is 1 meter, we can calculate the moment of inertia:
I_wood = (1/3) * 0.34 kg * (1 m)^2 = 0.113 kg*m^2.
We can now substitute the values into the formula for angular frequency:
ω_wood = sqrt(m * g * L / I_wood) = sqrt(0.34 kg * 9.8 m/s^2 * 1 m / 0.113 kg*m^2) = 5.19 rad/s.
Finally, to find the period, T_wood, we can use the formula: T = 2π/ω.
T_wood = 2π / ω_wood = 2π / 5.19 rad/s ≈ 1.21 seconds.
Therefore, the period of the wood pendulum is approximately 1.21 seconds.
(b) For the metal pendulum:
Given that the mass of the metal pendulum is 0.63 kg and the length of the meter stick is 1 meter, we can calculate the moment of inertia:
I_metal = (1/3) * 0.63 kg * (1 m)^2 = 0.210 kg*m^2.
Using the formula for angular frequency:
ω_metal = sqrt(m * g * L / I_metal) = sqrt(0.63 kg * 9.8 m/s^2 * 1 m / 0.210 kg*m^2) = 5.94 rad/s.
To find the period, T_metal, we use the formula: T = 2π/ω.
T_metal = 2π / ω_metal = 2π / 5.94 rad/s ≈ 1.06 seconds.
Therefore, the period of the metal pendulum is approximately 1.06 seconds.
To determine the periods of the wood and metal pendulums, we need to find the moments of inertia of the meter sticks. The formula for the moment of inertia of a thin rod rotating about one end, like in this case, is given by:
I = (1/3) * mL^2
Where m is the mass of the rod and L is its length.
For the wood pendulum, the mass is 0.34 kg and the length of the meter stick is 1 meter. So we can calculate its moment of inertia:
I_wood = (1/3) * 0.34 kg * (1 m)^2 = 0.1133 kg.m^2
Similarly, for the metal pendulum, the mass is 0.63 kg and the length of the meter stick is also 1 meter. We can calculate its moment of inertia:
I_metal = (1/3) * 0.63 kg * (1 m)^2 = 0.21 kg.m^2
Now that we have the moments of inertia, we can calculate the angular frequencies (omega) and then the frequencies (f) of the pendulums using the formula:
omega = sqrt(g/L * I)
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
For the wood pendulum:
omega_wood = sqrt(9.8 m/s^2 / 1 m * 0.1133 kg.m^2) = 3.55 rad/s
For the metal pendulum:
omega_metal = sqrt(9.8 m/s^2 / 1 m * 0.21 kg.m^2) = 4.13 rad/s
Finally, we can calculate the periods (T) using the formula:
T = 2*pi / omega
For the wood pendulum:
T_wood = 2*pi / 3.55 rad/s = 1.78 s
For the metal pendulum:
T_metal = 2*pi / 4.13 rad/s = 1.52 s
Therefore, the period of the wood pendulum is 1.78 seconds and the period of the metal pendulum is 1.52 seconds.
They are both meter sticks. Note that M/I appears in the frequency equation. I is proportional to M so Mass cancels out.
They will both have the same period.
If they swing from one end, I = M L^2/3
M/I = L^2/3
Maybe I don't understand..
If I=m (L^2) /3, then we have:
w=2 pi f = sqrt(mgL/I)
Subbing in I, we have:
2 pi f = sqrt(g L / (L^2/3) )
=sqrt(3g/L)
Thus, we have:
f=1/(2pi) * sqrt(3*g/L)=1/(2pi) * sqrt(3*9.8/1) = 0.862992.
This would mean that the period should be:
1/ freq = 1/0.862992, but I don't believe this answer is correct.
Can you please clarify?