A vertical spring A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx.
a. How much work is required to compress the spring and lift the mass 0.5 m?

Verified step by step guidance

1. Identify the forces and displacements involved: The mass stretches the spring 2 m from its equilibrium position, so the spring constant \( k \) can be found using Hooke's Law \( F = kx \), where \( F \) is the weight of the mass and \( x \) is the displacement.
2. Calculate the spring constant \( k \) by setting the force equal to the weight of the mass: \( F = mg = kx \), where \( m = 10 \) kg, \( g = 9.8 \) m/s\^2, and \( x = 2 \) m. Rearrange to find \( k = \frac{mg}{x} \).
3. Determine the total displacement for the work calculation: compressing the spring and lifting the mass by 0.5 m means the spring is compressed from its stretched position by 0.5 m, so the new displacement from equilibrium is \( 2 - 0.5 = 1.5 \) m.
4. Set up the integral for the work done in compressing the spring from 2 m to 1.5 m using the formula for work done by a variable force: \( W = \int_{x_1}^{x_2} F(x) \, dx = \int_{1.5}^{2} kx \, dx \).
5. Evaluate the integral to find the work done: integrate \( kx \) with respect to \( x \) over the limits from 1.5 to 2, which gives \( W = \left[ \frac{kx^2}{2} \right]_{1.5}^{2} \). This expression represents the work required to compress the spring and lift the mass.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a linear spring is proportional to its displacement from equilibrium, expressed as F(x) = kx, where k is the spring constant and x is the displacement. This relationship is fundamental for calculating forces and work involving springs.

Work Done by a Variable Force

When a force varies with position, the work done is found by integrating the force over the displacement. For a spring, work is calculated as the integral of F(x) = kx from the initial to final position, representing the energy required to compress or stretch the spring.

Gravitational Potential Energy

Lifting a mass against gravity increases its gravitational potential energy, calculated as mgh, where m is mass, g is acceleration due to gravity, and h is height. This energy must be included when determining the total work done in lifting the mass.