Problem 6 - Physics

Question

In a room that is \(2.44 \mathrm{~m}\) high, a spring (unstrained length \(=0.30 \mathrm{~m}\) ) hangs from the ceiling. A board whose length is \(1.98 \mathrm{~m}\) is attached to the free end of the spring. The board hangs straight down, so that its \(1.98-\mathrm{m}\) length is perpendicular to the floor. The weight of the board ( 104 N) stretches the spring so that the lower end of the board just extends to, but does not touch, the floor. What is the spring constant of the spring?

Step-by-Step Solution

Answer

The spring constant is 650 N/m.

Step 1: Understand the Physical Setup

The room is 2.44 m high. A spring with an unstrained length of 0.30 m hangs from the ceiling. A board of length 1.98 m is attached to the spring and hangs down towards the floor without touching it. The stretched system completes the room height but does not touch the floor.

Step 2: Determine the Total Extension Needed

The unstrained spring length plus the board length equals the height without stretching. Thus, the extension needed is from 2.44 m (room height) minus these lengths: \(2.44\, \text{m} - 0.30\, \text{m} - 1.98\, \text{m} = 0.16\, \text{m}\). Thus, the spring extends by 0.16 m.

Step 3: Understand Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to its extension: \( F = kx \), where \( F \) is the force in newtons, \( k \) is the spring constant in newton per meter (N/m), and \( x \) is the extension in meters.

Step 4: Apply Hooke's Law to Find the Spring Constant

Using Hooke's Law, and knowing that the force exerted by the board is its weight, 104 N, we can solve for the spring constant: \( k = \frac{F}{x} = \frac{104\, \text{N}}{0.16\, \text{m}} \).

Step 5: Calculate the Spring Constant

Perform the division to find \( k \): \( k = \frac{104}{0.16} = 650 \). Thus, the spring constant is 650 N/m.

Key Concepts

Spring Constant

The spring constant, represented by the symbol \( k \), is a measure of a spring's stiffness. It defines the relationship between the force applied to a spring and the distance it stretches or compresses. A higher spring constant means a stiffer spring, which requires more force to compress or extend by a specific amount.
The formula used to calculate the spring constant is derived from Hooke’s Law:

• \( F = kx \) where \( F \) is the force applied, \( x \) is the displacement from the spring's original length, and \( k \) is the spring constant in newtons per meter (N/m).

The spring constant is calculated by dividing the force applied by the extension (or compression) produced. In this exercise, the weight of the board (104 N) stretches the spring by 0.16 m, leading to a spring constant of 650 N/m.

Mechanical Equilibrium

Mechanical equilibrium refers to a state where all the forces acting on an object are balanced, resulting in no net force and therefore no acceleration. In other words, the object remains at rest or moves with constant velocity. When analyzing spring systems, ensuring mechanical equilibrium is crucial.
In this context, the board attached to the spring is in mechanical equilibrium when the force of gravity pulling it down is exactly matched by the spring force pulling it back up. This balance is what makes the board hover just above the floor without touching it.

• The gravitational force (104 N) exerted by the board is balanced by the upward spring force.

Mechanical equilibrium is an essential concept in understanding how forces interact to maintain stability in various physical systems.

Spring Force

Spring force is the force exerted by a spring when it is compressed or extended. This force acts in the opposite direction of the displacement. It is the mechanism that allows springs to store and release energy.
According to Hooke’s Law:

• The spring force \( F_s \) is given by \( F_s = kx \), where \( k \) is the spring constant and \( x \) is the displacement.

In this problem, the spring force is the upward force that balances the downward gravitational pull of the board. This force is crucial for ensuring mechanical equilibrium. Without it, the board would fall to the floor, indicating how spring force plays a vital role in maintaining balance in suspended systems.

Elasticity

Elasticity is the property of materials that enables them to return to their original shape after being deformed by an external force. Springs perfectly demonstrate elasticity by compressing or extending when a force is applied and then returning to their original length when the force is removed.
This concept is described numerically by Hooke’s Law, which reflects the proportional relationship between the force applied to a spring and its resultant displacement, within the elastic limit.

• The elastic limit is the maximum extent to which a material can be stretched without permanently altering its shape.
• Beyond the elastic limit, the material may not return to its original form when the force is removed.

Understanding elasticity helps explain why springs are used in various applications, from mechanical watches to car suspensions, providing efficient energy transfer and buffering capabilities.