Problem 33 - College Physics

Question

\(\bullet\) \(\bullet\) A 2.50 -kg mass is pushed against a horizontal spring of force constant 25.0 \(\mathrm{N} / \mathrm{cm}\) on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 11.5 \(\mathrm{J}\) of potential energy in it, the mass is sud- denly released from rest. (a) Find the greatest speed the mass reaches. When does this occur? (b) What is the greatest accel- eration of the mass, and when does it occur?

Step-by-Step Solution

Answer

(a) Max speed is 3.02 m/s when spring is fully extended; (b) Max acceleration is 96 m/s² at full compression.

Step 1: Understand Spring Force Formula

The potential energy (PE) stored in a spring is given by the formula: \[ \text{Potential Energy} = \frac{1}{2} k x^2 \]where \( k \) is the spring constant and \( x \) is the compression distance. We need this to find \( x \).

Step 2: Convert Units for Spring Constant

The spring constant is given as 25.0 N/cm. We need to convert this to N/m for consistency with standard SI units. \[ 25.0 \frac{\text{N}}{\text{cm}} = 2500 \frac{\text{N}}{\text{m}} \]

Step 3: Solve for Compression Distance

Using the potential energy equation and solving for \( x \):\[ 11.5 = \frac{1}{2} \cdot 2500 \cdot x^2 \]Simplifying this gives, \[ x^2 = \frac{11.5 \times 2}{2500} \]\[ x \approx 0.096 \text{ m} \]

Step 4: Apply Conservation of Energy

When the spring releases the mass, all stored potential energy is converted to kinetic energy (KE) at max speed. Thus, \[ 11.5 = \frac{1}{2} m v^2 \]

Step 5: Solve for Maximum Speed

Rearrange to solve for velocity \( v \):\[ v = \sqrt{\frac{2 \times 11.5}{2.50}} \approx 3.02 \text{ m/s} \]

Step 6: Determine Occurrence of Maximum Speed

The mass reaches its maximum speed the moment it is released and has traveled the entire compression distance of the spring (i.e., when the spring returns to its uncompressed state).

Step 7: Use Hooke's Law for Maximum Acceleration

Maximum acceleration occurs when force is highest at maximum compression:\[ F = k x \]\[ a = \frac{F}{m} = \frac{k x}{m} \]

Step 8: Calculate Maximum Acceleration

Substitute known values into acceleration formula:\[ a = \frac{2500 \times 0.096}{2.50} \approx 96 \text{ m/s}^2 \]

Step 9: Determine Occurrence of Maximum Acceleration

The greatest acceleration occurs at maximum compression, just before the mass starts to move.

Key Concepts

Potential Energy

Potential energy in the context of spring mechanics refers to the energy stored within the spring when it is compressed or stretched. This energy is given by the formula \[\text{Potential Energy} = \frac{1}{2} k x^2\]where \(k\) is the spring constant, indicating the stiffness of the spring, and \(x\) is the displacement from its equilibrium position. In this problem, the potential energy stored as the spring is compressed represents the energy available to do work once the mass is released. It is key to note that this stored energy will transform into kinetic energy as the spring returns to its original length, moving the mass in the process.

Conservation of Energy

The principle of conservation of energy in spring mechanics states that energy cannot be created or destroyed but only transformed from one form to another. This concept is crucial when analyzing systems involving springs. In our exercise, the potential energy stored in the spring is transformed entirely into kinetic energy when the spring returns to its normal, uncompressed state. This exchange can be mathematically understood by setting the initial potential energy equal to the kinetic energy of the mass:\[11.5 = \frac{1}{2} m v^2\]Where all initial potential energy is converted to the kinetic form, revealing that energy shifts efficiently between these states without any losses due to friction.

Hooke's Law

Hooke’s Law is foundational in understanding spring mechanics. It provides the relationship between the force exerted by a spring and its displacement. Stated mathematically, it is:\[F = k x\]In this formula, \(F\) is the force exerted by the spring, \(k\) is the spring constant, and \(x\) is the displacement from the equilibrium position. For the spring in this exercise, understanding Hooke's Law helps calculate the maximum force exerted and consequently, the maximum acceleration as the mass is pushed away from the spring. Essentially, Hooke's Law is used to understand when the spring is at its most compressed state which corresponds to maximum force and acceleration.

Kinetic Energy

Kinetic energy represents the energy of motion possessed by the object, in this case, the 2.50 kg mass once released from the spring. After the spring releases the stored potential energy, this energy is converted into kinetic energy, causing the mass to move. The formula for kinetic energy is given by:\[\text{Kinetic Energy} = \frac{1}{2} m v^2\]Where \(m\) is the mass of the object and \(v\) is its velocity. The maximum speed of the mass occurs when all the potential energy has been converted to kinetic energy, showing a direct relationship between stored potential energy and the velocity achieved by the mass.

Spring Constant

The spring constant \(k\) is a measure of a spring's stiffness or resistance to being compressed or stretched. In our exercise, the given spring constant is 2500 N/m, converted from 25 N/cm for better consistency with standard SI units. The spring constant plays a significant role in calculations involving both potential energy stored and forces exerted by the spring.

• High \(k\) value: indicates a very stiff spring, requiring more force for the same displacement.
• Low \(k\) value: indicates a more flexible spring, requiring less force for the same displacement.

Understanding the spring constant is essential to compute how much a spring can be compressed, the energy it stores, and the maximum forces it can exert during its operation.