Mechanics (M2) - Work, Energy and Power: Further Applications, Elastic Strings and Springs

1. Work, Energy and Power: Further Applications

In this section, we extend the concepts of work, energy, and power to more complex scenarios. This includes dealing with non-uniform forces and understanding the work-energy theorem in greater detail.

1.1 Non-Uniform Forces

When a body moves under a non-uniform force, the work done is not simply the area under the force-displacement graph. We need to calculate the work done over small displacements and then integrate.

Consider a force F(x) acting on an object moving along the x-axis. The work done in moving the object from x = a to x = b is given by:

$$W = \int_{a}^{b} F(x) dx$$

This integral represents the area under the F(x) curve between x = a and x = b.

1.2 Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.

$$W_{net} = \Delta KE = KE_f - KE_i$$

Where Wnet is the net work done, KEf is the final kinetic energy, and KEi is the initial kinetic energy.

This theorem is particularly useful when the forces involved are not constant.

1.3 Potential Energy and Conservative Forces

For conservative forces (like gravity or spring forces), the work done in moving an object between two points is independent of the path taken. This allows us to define potential energy.

The potential energy U associated with a conservative force is defined as:

$$U = -\int F(x) dx$$

The change in potential energy is equal to the negative of the work done by the conservative force:

$$\Delta U = -W_{conservative}$$

2. Elastic Strings and Springs

Elastic strings and springs obey Hooke's Law, which relates the force exerted by the material to the extension or compression.

2.1 Hooke's Law

Hooke's Law states that the force exerted by an elastic string or spring is proportional to the displacement from its equilibrium position.

$$F = -kx$$

Where F is the force, k is the spring constant (a measure of stiffness), and x is the displacement from equilibrium.

The negative sign indicates that the force is in the opposite direction to the displacement.

2.2 Potential Energy of a Spring

The potential energy stored in a spring or elastic string is given by:

$$U = \frac{1}{2}kx^2$$

This potential energy is stored when the spring is stretched or compressed.

2.3 Applications of Hooke's Law

Hooke's Law is used to determine the spring constant k of a spring experimentally. By measuring the extension or compression x for known forces F, we can calculate k.

For a string, the potential energy is also given by:

$$U = \frac{1}{2}m\omega^2 y^2$$

Where m is the mass of the string and y is the displacement.

2.4 Combined Springs

When multiple springs are connected in series or parallel, their effective spring constant is determined by the arrangement.

Series Connection: The effective spring constant keq is given by:

$$\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + \dots + \frac{1}{k_n}$$

Parallel Connection: The effective spring constant keq is given by:

$$\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + \dots + \frac{1}{k_n}$$

The effective spring constant is the sum of the individual spring constants.

Concept	Formula	Description
Work Done by a Non-Uniform Force	$$W = \int_{a}^{b} F(x) dx$$	Area under the force-displacement curve.
Work-Energy Theorem	$$W_{net} = \Delta KE = KE_f - KE_i$$	Net work done equals the change in kinetic energy.
Potential Energy of a Spring	$$U = \frac{1}{2}kx^2$$	Energy stored in a stretched or compressed spring.
Effective Spring Constant (Series)	$$\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + \dots + \frac{1}{k_n}$$	Springs connected end-to-end.
Effective Spring Constant (Parallel)	$$\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + \dots + \frac{1}{k_n}$$	Springs connected side-by-side.