Problem 54 - University Physics with Modern Physics

Question

A 1.80-kg monkey wrench is pivoted 0.250 \(\mathrm{m}\) from its center of mass and allowed to swing as a physical pendulum. The period for small-angle oscillations is 0.940 s. (a) What is the moment of inertia of the wrench about an axis through the pivot? (b) If the wrench is initially displaced 0.400 rad from its equilibrium position, what is the angular speed of the wrench as it passes through the equilibrium position?

Step-by-Step Solution

Answer

(a) The moment of inertia is approximately 0.0745 kg·m². (b) The angular speed is approximately 1.287 rad/s.

Step 1: Calculating the Moment of Inertia

To find the moment of inertia, use the formula for the period of a physical pendulum:\[ T = 2\pi \sqrt{\frac{I}{mgd}} \]where \(T = 0.940\, s\), \(m = 1.80\, kg\), \(g = 9.81\, m/s^2\), and \(d = 0.250\, m\).Rearrange the equation to solve for \(I\):\[ I = \frac{T^2 \cdot mgd}{4\pi^2} \]Substitute the known values:\[ I = \frac{(0.940)^2 \cdot 1.80 \cdot 9.81 \cdot 0.250}{4\pi^2} \]Calculate \(I\):\[ I \approx 0.0745\, \text{kg}\cdot \text{m}^2 \]

Step 2: Finding the Angular Speed at Equilibrium

When the wrench is initially displaced by \(\theta_0 = 0.400\, \text{rad}\), use conservation of mechanical energy:Potential energy at the top:\[ PE = mgh_{\text{max}} \]where \(h_{\text{max}} = d(1 - \cos\theta_0)\).Kinetic energy at equilibrium (where \(v = \omega d\)):\[ KE = \frac{1}{2} I \omega^2 \]Set \(PE = KE\):\[ mgd(1 - \cos\theta_0) = \frac{1}{2} I \omega^2 \]Rearrange to solve for \(\omega\):\[ \omega = \sqrt{\frac{2mgd(1 - \cos\theta_0)}{I}} \]Substitute known values:\[ \omega = \sqrt{\frac{2 \cdot 1.80 \cdot 9.81 \cdot 0.250 \cdot (1 - \cos(0.400))}{0.0745}} \]\[ \omega \approx 1.287\, \text{rad/s} \]

Key Concepts

Moment of Inertia

Understanding the moment of inertia is crucial when analyzing rotational motions, such as those found in physical pendulums. It essentially describes how difficult it is to change the rotational motion of an object. For a monkey wrench swinging about a pivot, the moment of inertia (I) depends on the mass distribution relative to the axis of rotation. The formula for calculating the moment of inertia in this context is derived from the period (T) equation of a physical pendulum:\[ T = 2\pi \sqrt{\frac{I}{mgd}} \]where

• I is the moment of inertia,
• m is the mass of the object (1.80 kg for the wrench),
• g is the acceleration due to gravity (9.81 m/s²),
• d is the distance from the pivot to the center of mass (0.250 m).

By rearranging the equation, we can solve for I and input the given values to find I \, \approx \, 0.0745 \, \text{kg} \cdot \text{m}^2. This tells us how much rotational inertia the wrench exhibits as it swings.

Angular Speed

Angular speed (\omega) signifies how fast an object rotates around a pivot in radians per second. In the case of the swinging wrench, angular speed varies depending on its position in its swing. When the wrench is released from a displaced position, it accelerates due to gravitational forces.Using the conservation of energy principle, we know that the wrench will convert potential energy at its highest point into kinetic energy at its equilibrium position (lowest point). The expressions for potential energy at the highest point and kinetic energy are:Potential Energy: \[ PE = mgh_{max} \]Kinetic Energy: \[ KE = \frac{1}{2} I \omega^2 \]where \( h_{max} = d(1 - \cos\theta_0) \) and\( \theta_0 \) is the initial displacement. By setting the initial potential energy equal to the kinetic energy at equilibrium, we can solve for \omega, yielding approximately 1.287 rad/s.

Conservation of Energy

The conservation of energy is a foundational principle in physics, stating that energy cannot be created or destroyed, only transformed from one form to another. In the context of a physical pendulum like a monkey wrench, this involves the transformation between potential and kinetic energy.At the highest point in its swing, the pendulum has maximum potential energy and zero kinetic energy. As it swings down to the equilibrium position, potential energy decreases while kinetic energy increases. At the lowest point, all potential energy is converted to kinetic energy.For calculations:

• Potential Energy (PE) at the highest point: \( PE = mgh_{max} \).
• Kinetic Energy (KE) at equilibrium: \( KE = \frac{1}{2}I\omega^2 \).

By equating PE and KE, we can solve for unknown variables, such as the angular speed (\omega) when potential energy is wholly transformed into kinetic energy.

Small-angle Oscillations

In physics, small-angle oscillations describe conditions where the angle of displacement is small enough (\theta \approx 10^{\circ} or less) that certain simplifying assumptions can be made. When this condition is met, the restoring force in pendulum motion is proportional to the angle of displacement, leading to simple harmonic motion.For the monkey wrench, a displacement of 0.400 rad is relatively small, allowing us to use the small-angle approximation: \( \sin\theta \approx \theta \) . This simplifies computations in formulas used to describe the motion, such as oscillation period or energy equations.The period of a pendulum exhibiting small-angle oscillations is given by the formula: \[ T = 2\pi\sqrt{\frac{I}{mgd}} \]used to find the moment of inertia. This approximation helps in deriving accurate solutions for systems behaving like simple harmonic oscillators.