Problem 15 - College Physics

Question

\(\bullet\bullet\) A 0.25 -kg object is suspended on a light spring of spring constant \(49 \mathrm{~N} / \mathrm{m}\) and the system is allowed to come to rest at its equilibrium position. The object is then pulled down \(0.10 \mathrm{~m}\) from the equilibrium position and released. What is the speed of the object when it goes through the equilibrium position?

Step-by-Step Solution

Answer

The speed of the object at the equilibrium position is approximately 1.4 m/s.

Step 1: Identify Known Quantities

We are given an object's mass, \( m = 0.25 \, \text{kg} \), and the spring constant, \( k = 49 \, \text{N/m} \). Additionally, the displacement from equilibrium is \( x_0 = 0.10 \, \text{m} \).

Step 2: Use Conservation of Energy

The problem can be solved using the conservation of mechanical energy. At the point of release, the energy is all potential: \( U_i = \frac{1}{2} k x_0^2 \). At the equilibrium position, energy will all be kinetic: \( K_f = \frac{1}{2} m v^2 \).

Step 3: Solve for Kinetic Energy at Equilibrium

Set the initial potential energy equal to the kinetic energy at equilibrium: \( \frac{1}{2} k x_0^2 = \frac{1}{2} m v^2 \).

Step 4: Substitute Known Values

Plug in the values: \( \frac{1}{2} (49) (0.10)^2 = \frac{1}{2} (0.25) v^2 \). Simplify to find \( v^2 = \frac{49 \times 0.01}{0.25} \).

Step 5: Calculate the Speed

Perform the division to get \( v^2 = 1.96 \). Take the square root of both sides to find \( v = \sqrt{1.96} \approx 1.4 \, \text{m/s} \).

Key Concepts

Harmonic Motion

Simple harmonic motion is a fascinating concept in physics that describes the back-and-forth oscillatory movement of an object, like a pendulum or spring, around an equilibrium position. Imagine a swinging pendulum or a mass at the end of a spring; these are classic examples of harmonic motion in action.
Here's how it works:

• An object in harmonic motion experiences a restoring force proportional to its displacement from its equilibrium position. This means that the further it is pulled away from this point, the stronger the force acting to bring it back.
• This type of motion is characterized by repeating cycles, known as oscillations, which can continue indefinitely in the absence of energy losses.
• The most common examples include swinging pendulums and vibrating springs.

Understanding simple harmonic motion is crucial because it forms the basis of more complex mechanical systems. It is also linked to energy conservation principles, as seen in this problem where total mechanical energy remains constant.

Spring Constant

The spring constant is a key value in understanding how a spring behaves. It indicates the stiffness of a spring, governing how much force is needed to compress or elongate it by a specific amount. In our problem, the spring constant is given as 49 N/m, which tells us the following:

• A spring with a high spring constant is stiff and requires more force to stretch or compress it. Conversely, a lower spring constant implies a softer spring.
• The spring constant appears in Hooke's Law: \[ F = kx \]where \(F\) is the force applied to the spring, \(k\) is the spring constant, and \(x\) is the displacement (stretch or compression).

This concept is essential for calculated movements in systems using springs, like in engineering and physics problems, where knowing the spring constant allows us to predict how a system will respond to forces.

Mechanical Energy

Mechanical energy in physics refers to the sum of potential energy and kinetic energy in a system. In the context of our spring problem, mechanical energy transforms between two states as the object oscillates:

• Potential Energy (PE): This is the energy stored in the system due to its position, such as the compressed spring in our problem. At the start, the object stores energy as potential energy: \[ U = \frac{1}{2} k x^2 \]where \(x\) is the displacement from equilibrium.
• Kinetic Energy (KE): This is the energy of motion. As the object passes through the equilibrium position, all potential energy converts into kinetic energy: \[ K = \frac{1}{2} mv^2 \]

The principle of conservation of mechanical energy tells us that in the absence of external forces like friction, the total mechanical energy (PE + KE) remains constant. This principle was used to find the speed of the object as it passed through equilibrium.