Question

The timing device shown consists of a movable cylinder of known mass \(m\) that is attached to a rod of negligible mass supported by a torsional spring at its base. If the stiffness of the spring is \(k_{T}\), where \(k_{T} / m g L>1\), determine the period of small angle motion of the device as a function of the attachment length, \(L\), if the spring is untorqued when \(\theta=0\).

Short answer

Answer: The period of small angle motion for the device is given by \(T = \frac{2\pi}{\sqrt{\frac{k_T + mgL}{I}}}\), where \(k_T\) is the stiffness of the torsional spring, \(m\) is the mass of the cylinder, \(g\) is the acceleration due to gravity, and \(I\) is the moment of inertia of the cylinder.

Step-by-step solution

Define the relevant variables and quantities

We are given the mass of the cylinder (\(m\)), the stiffness of the spring (\(k_T\)), and the attachment length (\(L\)). We are also given the condition that \(k_T / mgL > 1\). Our goal is to determine the period of motion (\(T\)) as a function of the attachment length, \(L\).

Draw a free body diagram and analyze the forces

Draw a free body diagram of the system, showing the torsional spring attached to the rod with the cylinder at the end. We can identify two forces acting on the system: gravitational force, \(mg\), acting vertically downwards on the cylinder, and the torque of the spring, \(-k_T\theta\).

Write the equation of motion for the cylinder

Now we need to write the equation of motion for the cylinder, which links its angular displacement, \(\theta\), to the forces and torques acting on the system. We will use the Newton's second law for rotation, which states that the net torque acting on the cylinder is equal to the moment of inertia times the angular acceleration: \(-k_T\theta - mgL \sin\theta = I \ddot{\theta}\), where \(I\) is the moment of inertia of the cylinder, and \(\ddot{\theta}\) is the angular acceleration.

Simplify the equation for small angles

Since we are interested in the period of small angle motion, we can use the small angle approximation: \(\sin\theta \approx \theta\). This allows us to simplify the equation of motion: \(-k_T\theta - mgL\theta = I \ddot{\theta}\).

Rewrite the equation as a standard simple harmonic motion equation

We can rewrite the previous equation as follows: \(\ddot{\theta}+\frac{k_T + mgL}{I}\theta = 0\). This equation is in the form of a standard simple harmonic motion equation, which can be written as: \(\ddot{\theta} + \omega^2 \theta = 0\), where \(\omega\) is the angular frequency. Comparing the two equations, we get: \(\omega^2 = \frac{k_T + mgL}{I}\).

Calculate the period T in terms of L

The period of simple harmonic motion, \(T\), is related to the angular frequency, \(\omega\), by the formula: \(T = \frac{2\pi}{\omega}\). Using the expression for \(\omega^2\), we can find the period: \(T = \frac{2\pi}{\sqrt{\frac{k_T + mgL}{I}}}\). Now we have the period of small angle motion of the device as a function of the attachment length, \(L\).

Key concepts

Understanding Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion or oscillation that takes place when the restoring force on an object is directly proportional to the object's displacement from its equilibrium position and is directed towards that equilibrium position. Imagine pulling a weight on a spring and then releasing it; the weight will oscillate back and forth. This is a classic example of simple harmonic motion.

In the context of our timing device exercise, the motion of the cylinder attached to the torsional spring can exhibit SHM under certain conditions. When the cylinder is displaced, the spring tries to bring it back to the initial position, resulting in an oscillatory motion.

Mathematically, SHM can be described by the equation \( \ddot{x} + \omega^2 x = 0 \) where \( \ddot{x} \) is the acceleration of the object, \( x \) is the displacement from equilibrium, and \( \omega \) is the angular frequency of the motion. The angular frequency is related to the physical properties of the system and the period of the motion by \( \omega = \frac{2\pi}{T} \) where \( T \) is the period of the oscillation. In the improvement of the exercise, it's critical to grasp that for small angles, the oscillatory motion of the system can be described using SHM principles.

Dynamics of a Torsional Spring

A torsional spring is a mechanical element that exhibits a twisting motion when torque is applied to it. When it comes to torsional springs, the restoring torque is similar to the restoring force in a linear spring; it is proportional to the angle of twist from its equilibrium position.

In the exercise's timing device, the torsional spring provides the restoring torque needed to create the oscillatory motion of the cylinder. The stiffness or rigidity of the spring, denoted by \( k_{T} \), plays a crucial role in determining how the system behaves when subjected to displacements from its equilibrium position.

The relationship between torque \( \tau \) and the angle of twist \( \theta \) in a torsional spring is given by Hooke's law for torsion: \( \tau = -k_{T}\theta \). When \( \theta \) is small, \( \sin\theta \) is approximately equal to \( \theta \) itself, which simplifies the analysis of the system's motion. The negative sign indicates that the torque tries to return the system to its equilibrium position, reflecting the property of a restoring torque.

Equation of Motion for Torsional Systems

The equation of motion describes how an object moves under the influence of forces. In rotational systems like our torsional spring and cylinder, the equation of motion links the angular displacement to the torques acting on the system.

To arrive at the equation of motion, we apply Newton's second law for rotation, which in this case results in \( -k_{T}\theta - mgL\sin\theta = I \ddot{\theta} \), where \( I \) represents the moment of inertia of the cylinder, and \( \ddot{\theta} \) is the angular acceleration. For small angles, we approximate \( \sin\theta \approx \theta \) to simplify the equation.

Following this simplification, we recognize the form of the equation that corresponds to simple harmonic motion. By comparing it to the standard SHM equation, \( \ddot{\theta} + \omega^2\theta = 0 \), we can extract an expression for the angular frequency \( \omega \) and ultimately the period of the motion \( T \).

Understanding the equation of motion is paramount because it provides a direct connection between the physical parameters of the system and the characteristics of the motion it experiences, such as its period. The step by step solution guides students through manipulating the equation to reveal insights about how the length of the attachment (\(L\)) influences the timing device's oscillation period.