Chapter 5: Problem 6

Evaluate the mechanical impedance and the mechanical admittance of the simple pendulum.

Short Answer

Expert verified

Answer: The formulas for the mechanical impedance (Z_m) and mechanical admittance (Y_m) of a simple pendulum are: $$Z_m = \left(m\frac{4\pi^2 l}{gT^2} - \frac{mg}{x}\right)$$ $$Y_m = \frac{1}{\left(m\frac{4\pi^2 l}{gT^2} - \frac{mg}{x}\right)}$$

Step by step solution

Determine the mass of the pendulum

Since we're given no specific information about the simple pendulum, let's assume we have a pendulum of mass $m$.

Determine the angular frequency

The angular frequency ($\omega$) of a simple pendulum can be derived from its period ($T$), which is given by: $$T = 2\pi\sqrt{\frac{l}{g}}$$ where $l$ is the length of the pendulum and $g$ is the acceleration due to gravity. To find the angular frequency, we can use the relationship: $$\omega = \frac{2\pi}{T}$$

Determine the spring constant

In a simple pendulum, the restoring force ($F$) is proportional to the displacement from the equilibrium position ($x$), giving rise to the spring constant ($k$). The proportionality can be expressed as: $$F = -kx$$ Since the restoring force is also equal to the gravitational force acting on the mass, we can write: $$mg = kx$$ Thus, the spring constant ($k$) can be expressed as: $$k = \frac{mg}{x}$$

Calculate the mechanical impedance

Now that we have the mass, angular frequency, and spring constant, we can calculate the mechanical impedance ($Z_m$) using the formula: $$Z_m = (m\omega^2 - k)$$ Substituting the expressions for $\omega$ and $k$ from steps 2 and 3, we have: $$Z_m = \left(m\frac{4\pi^2 l}{gT^2} - \frac{mg}{x}\right)$$

Calculate the mechanical admittance

To find the mechanical admittance ($Y_m$), we simply take the reciprocal of the mechanical impedance: $$Y_m = \frac{1}{Z_m}$$ Substituting the expression for $Z_m$ from step 4, we get: $$Y_m = \frac{1}{\left(m\frac{4\pi^2 l}{gT^2} - \frac{mg}{x}\right)}$$ In conclusion, the mechanical impedance ($Z_m$) and the mechanical admittance ($Y_m$) of the simple pendulum are given by the formulas: $$Z_m = \left(m\frac{4\pi^2 l}{gT^2} - \frac{mg}{x}\right)$$ $$Y_m = \frac{1}{\left(m\frac{4\pi^2 l}{gT^2} - \frac{mg}{x}\right)}$$

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Evaluate the mechanical impedance and the mechanical admittance of the simple pendulum.

Short Answer

Step by step solution

Determine the mass of the pendulum

Determine the angular frequency

Determine the spring constant

Calculate the mechanical impedance

Calculate the mechanical admittance

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