Problem 37 - University Physics with Modern Physics

Question

A \(175-\) glider on a horizontal, frictionless air track is attached to a fixed ideal spring with force constant 155 \(\mathrm{N} / \mathrm{m}\) . At the instant you make measurements on the glider, it is moving at 0.815 \(\mathrm{m} / \mathrm{s}\) and is 3.00 \(\mathrm{cm}\) from its equilibrium point. Use energy conservation to find (a) the amplitude of the motion and (b) the maximum speed of the glider. (c) What is the angular frequency of the oscillations?

Step-by-Step Solution

Answer

The amplitude is 3.14 cm, the maximum speed is 0.970 m/s, and the angular frequency is 29.8 rad/s.

Step 1: Understand the Problem

We need to find the amplitude of the motion, the maximum speed of the glider, and the angular frequency of the oscillations. The glider's mass is moving at a certain speed and is at a certain displacement from the equilibrium point. We will use the concepts of energy conservation and simple harmonic motion (SHM).

Step 2: Identify Given Information

The mass of the glider, \(m\), is 175 g or 0.175 kg (since we need it in kg). The spring constant, \(k\), is 155 N/m. The initial speed, \(v\), of the glider is 0.815 m/s, and the displacement from the equilibrium point, \(x\), is 3.00 cm or 0.03 m.

Step 3: Apply Conservation of Energy for Amplitude

Using the conservation of energy principle, the total mechanical energy is constant. It includes kinetic energy (\(KE\)) and potential energy (\(PE\)). At an arbitrary point, we have \(KE + PE = \frac{1}{2} mv^2 + \frac{1}{2} kx^2 = \frac{1}{2} kA^2\), where \(A\) is the amplitude.Calculate: \[\frac{1}{2} (0.175)(0.815)^2 + \frac{1}{2} (155)(0.03)^2 = \frac{1}{2} (155)A^2\]

Step 4: Solve for Amplitude

Rearrange the equation to solve for \(A\):\[A^2 = \frac{0.5 \times 0.175 \times 0.815^2 + 0.5 \times 155 \times 0.03^2}{0.5 \times 155} \]Calculate \(A\):\[A \approx 0.0314 \text{ m or } 3.14 \text{ cm}\]

Step 5: Calculate Maximum Speed

The maximum speed occurs when all the energy is kinetic, i.e., at the equilibrium point where potential energy is zero:\[\frac{1}{2}mv_{max}^2 = \frac{1}{2}kA^2\]Solve for \(v_{max}\):\[v_{max}^2 = \frac{kA^2}{m}\]\[v_{max} = \sqrt{\frac{155 \times 0.0314^2}{0.175}}\]\[v_{max} \approx 0.970 \text{ m/s}\]

Step 6: Determine Angular Frequency

The angular frequency \(\omega\) of an oscillating spring-mass system is given by:\[\omega = \sqrt{\frac{k}{m}}\]Calculate \(\omega\):\[\omega = \sqrt{\frac{155}{0.175}}\]\[\omega \approx 29.8 \, \text{rad/s}\]

Key Concepts

Energy Conservation

In physics, energy conservation is a crucial concept, especially in harmonic motion. It implies that within a closed system, the total mechanical energy remains unchanged over time. This energy is comprised of two components: kinetic energy (energy due to motion) and potential energy (energy stored in the spring). In the context of harmonic motion with a spring and a mass, the total energy can be expressed as:

• Kinetic Energy, \( KE = \frac{1}{2}mv^2 \)
• Potential Energy, \( PE = \frac{1}{2}kx^2 \)

For a glider attached to a spring, energy conservation means that the sum of these energies at any point in time is equal to the maximum potential energy stored when the spring is at its maximum compression or elongation. Hence, the energy equation for this system is given by:
\[ KE + PE = \frac{1}{2} kA^2 \]Where:

• \(A\) is the amplitude of the oscillation
• \(k\) is the spring constant

By using this equation, you can find the amplitude when given the mass, velocity, and displacement of the glider.

Spring Constant

The spring constant, denoted by \(k\), is a measure of the stiffness of a spring. It indicates how much force is needed to extend or compress the spring by a unit length. The spring constant is expressed in units of \( \, \mathrm{N}/\mathrm{m} \).
A larger spring constant means a stiffer spring, which requires more force for the same amount of stretch compared to a spring with a smaller constant. In our problem, the spring constant is 155 \(\mathrm{N}/\mathrm{m}\). This means:

• For every meter the spring is stretched (or compressed), it needs a force of 155 newtons.

The spring constant is crucial for calculating the potential energy stored in a spring and for determining the angular frequency in simple harmonic motion. It directly affects the amplitude of the oscillations and the maximum speed of the object attached.

Oscillations

Oscillations refer to the repeated back and forth motion of an object. In this context, it is the motion of the glider as it moves around the equilibrium point when attached to a spring. Such movement is periodic and can be described through simple harmonic motion (SHM). In SHM:

• The amplitude is the maximum displacement from the equilibrium position.
• The motion is sinusoidal, meaning it can be described by sine and cosine functions.

Each complete cycle of oscillation involves movement from one position, through the equilibrium, to the opposite extreme, and back again. This type of motion is key in calculating elements like the glider's maximum speed and amplitude based on conservation of energy. Understanding oscillations allows us to predict the motion behavior of objects in systems like pendulums or springs.

Angular Frequency

Angular frequency, denoted by \(\omega\), is an important parameter in oscillatory motion, measured in radians per second (rad/s). It represents the rate of rotation. In the context of spring-mass systems, angular frequency relates to how quickly an object oscillates back and forth.
The formula for angular frequency in a spring-mass system is given by:\[ \omega = \sqrt{\frac{k}{m}} \]Where:

• \(k\) is the spring constant
• \(m\) is the mass of the object

With a higher angular frequency, the object completes more oscillations in a given time. In this problem, the calculated angular frequency tells us how fast the glider oscillates along the track. Understanding \(\omega\) is essential for grasping the dynamics of oscillatory systems and is fundamental in designing systems where timing of oscillations matters, like in clocks or measuring devices.