Problem 7 - College Physics

Question

\(\bullet\) An object of mass \(1.0 \mathrm{~kg}\) is attached to a spring with spring constant \(15 \mathrm{~N} / \mathrm{m}\). If the object has a maximum speed of \(0.50 \mathrm{~m} / \mathrm{s}\), what is the amplitude of oscillation?

Step-by-Step Solution

Answer

The amplitude of oscillation is approximately 0.129 m.

Step 1: Understand the Problem

We need to find the amplitude of oscillation for an object attached to a spring with mass \(1.0\, \text{kg}\) and a spring constant of \(15\, \text{N/m}\), given that its maximum speed is \(0.50\, \text{m/s}\).

Step 2: Use Energy Conservation Principle

The maximum kinetic energy the object can have is when the spring is at its natural length (i.e., the speed is maximum, and potential energy is zero). The total mechanical energy of the system (sum of kinetic and potential energy) is conserved.

Step 3: Write Down the Kinetic Energy Formula

The kinetic energy is given by the formula \( KE = \frac{1}{2}mv^2 \). Substitute the mass \( m = 1.0 \) kg and velocity \( v = 0.50 \) m/s.

Step 4: Calculate Maximum Kinetic Energy

Calculate the maximum kinetic energy:\[ KE = \frac{1}{2} \times 1.0 \times (0.50)^2 = \frac{1}{2} \times 0.25 = 0.125 \, \text{Joules} \]

Step 5: Set Total Energy Equal to Maximum Potential Energy

The total mechanical energy is equal to the maximum potential energy when the displacement is at the amplitude (maximum displacement).\[ E = \frac{1}{2} k A^2 \]Here, \( E \) is the same as the maximum kinetic energy.

Step 6: Solve for Amplitude A

Equate the maximum kinetic energy to the maximum potential energy:\[ 0.125 = \frac{1}{2} \times 15 \times A^2 \]Solve for \( A \):\[ A^2 = \frac{0.125 \times 2}{15} = \frac{0.25}{15} = \frac{1}{60} \]\[ A = \sqrt{\frac{1}{60}} \approx 0.129 \text{ m} \]

Step 7: Conclusion

The calculated amplitude of the oscillation is approximately \( 0.129 \) meters.

Key Concepts

Spring Constant

When dealing with the oscillation of a mass attached to a spring, the spring constant is a crucial concept to understand. It is denoted by the symbol \( k \) and expressed in units of \( ext{N/m} \) (newtons per meter). The spring constant measures the stiffness of the spring. In simpler terms, it tells us how much force is needed to stretch or compress the spring by one meter.

• A higher spring constant means a stiffer spring, requiring more force for the same amount of stretch.
• A lower spring constant indicates a more flexible spring, needing less force for the same displacement.

In our exercise, the spring constant is given as \( 15 ext{ N/m} \). This value helps determine how the spring behaves when a force is applied and is essential for calculating the amplitude of oscillation. It's part of the formula for potential energy, which is related to the stretch or compression of the spring.

Maximum Speed

The maximum speed of an object attached to a spring is an interesting point to explore. At maximum speed, the object is usually at the equilibrium position of the spring. At this position, all the stored potential energy in the spring is converted into kinetic energy.

• The speed is highest when the spring is neither compressed nor stretched, meaning potential energy is zero.
• The entire mechanical energy is then represented by the object’s kinetic energy.

For our specific exercise, the maximum speed is \( 0.50 ext{ m/s} \). This speed is used in the kinetic energy formula \( KE = \frac{1}{2} m v^2 \) to find out how much kinetic energy the object possesses. This value is crucial for understanding energy distribution during the oscillation.

Mass of Object

The mass of the object is a fundamental part of many physics equations involving motion and oscillation. It is commonly denoted by the symbol \( m \) and in this exercise, the mass is \( 1.0 ext{ kg} \). Mass plays an essential role in determining:

• The amount of kinetic energy as it is directly proportional to both mass and the square of the velocity \( KE = \frac{1}{2} m v^2 \).
• The motion dynamics when combined with other forces or constants like the spring constant \( k \).

Knowing the mass allows us to calculate how much energy is involved in the motion and how the object will behave under certain forces. In spring oscillations, this mass affects how fast the object moves and how it impacts the spring's potential energy.

Energy Conservation

Energy conservation is a powerful principle in understanding oscillation. It states that energy in an isolated system remains constant; it can neither be created nor destroyed but only transformed from one form to another. In spring-mass systems:

• The total mechanical energy is the sum of potential energy (stored energy in the spring) and kinetic energy (energy of motion).
• At maximum stretch or compression, energy is fully potential. At maximum speed, energy is fully kinetic.

For this exercise, using energy conservation, we equate the maximum kinetic energy to the maximum potential energy. This approach allows us to calculate the amplitude of oscillation, which is the maximum displacement from the equilibrium. By solving the equation \( 0.125 = \frac{1}{2} \, 15 \, A^2 \), we deduce the amplitude using energy conservation, helping us understand energy distribution during oscillation.