A 35-cm-long vertical spring has one end fixed on the floor. Placing a 2.2 kg physics textbook on the spring compresses it to a length of 29 cm. What is the spring constant?

Verified step by step guidance

1. Identify the given values: The original length of the spring is 35 cm (0.35 m), and the compressed length is 29 cm (0.29 m). The mass of the textbook is 2.2 kg. The force exerted by the textbook on the spring is due to gravity, which is calculated as \( F = m \cdot g \), where \( g \) is the acceleration due to gravity (approximately 9.8 m/s²).
2. Calculate the compression of the spring: The compression \( \Delta x \) is the difference between the original length and the compressed length, \( \Delta x = 0.35 \ \text{m} - 0.29 \ \text{m} \).
3. Write down Hooke's Law: Hooke's Law states that the force exerted by a spring is proportional to its compression, \( F = k \cdot \Delta x \), where \( k \) is the spring constant and \( \Delta x \) is the compression.
4. Rearrange Hooke's Law to solve for the spring constant \( k \): \( k = \frac{F}{\Delta x} \). Substitute \( F = m \cdot g \) into the equation, so \( k = \frac{m \cdot g}{\Delta x} \).
5. Substitute the known values into the equation: Use \( m = 2.2 \ \text{kg} \), \( g = 9.8 \ \text{m/s}^2 \), and \( \Delta x \) (calculated in step 2) to find the spring constant \( k \).

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position, expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. This principle is fundamental in understanding how springs behave under load.

Spring Constant (k)

The spring constant, denoted as k, quantifies the stiffness of a spring. A higher value of k indicates a stiffer spring that requires more force to compress or extend. It is calculated using the formula k = F/x, where F is the force applied and x is the displacement from the spring's natural length.

Gravitational Force

Gravitational force is the attractive force between two masses, calculated using the formula F = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.81 m/s² on Earth). In this context, the weight of the textbook compresses the spring, providing the force needed to apply Hooke's Law.