week 7 handout

Elastic Potential Energy and Oscillatory Systems

Elastic Potential Energy of a Spring

Springs are classic examples of systems that store elastic potential energy when compressed or stretched from their equilibrium position. This energy can be transformed into kinetic energy as the spring returns to equilibrium.

• Definition: Elastic potential energy is the energy stored in an object when it is deformed (stretched or compressed).

• Formula: The potential energy stored in a spring displaced by a distance $x$ from its equilibrium position $x_{eq}$ is given by: $U_{sp}(x) = \frac{1}{2}k(x - x_{eq})^2$ where k is the spring constant (a measure of the spring's stiffness).

• Graphical Representation: The energy vs. displacement graph for a spring is a parabola, showing that energy increases quadratically with displacement.

• Example: Stretching a spring by 0.1 m with $k = 200$ N/m stores $U_{sp} = \frac{1}{2} \times 200 \times (0.1)^2 = 1$ Joule.

Energy Exchange in Oscillatory Systems

Oscillators, such as springs and pendulums, exhibit periodic exchange between potential energy and kinetic energy. The total mechanical energy remains constant (in the absence of friction).

• Key Point: As the system oscillates, potential energy is converted to kinetic energy and vice versa.

• Graphical Representation:

  • Potential energy (PE) and kinetic energy (KE) are out of phase: when PE is maximum, KE is minimum, and vice versa.

  • Energy vs. time graphs for both PE and KE are sinusoidal and mirror each other.

• Example: In a mass-spring system, when the mass passes through equilibrium, KE is maximum and PE is zero; at maximum displacement, PE is maximum and KE is zero.

Pendulum: Kinetic and Potential Energy Over Time

A pendulum is another example of an oscillating system. As the pendulum swings, its energy alternates between kinetic and gravitational potential energy.

• Potential Energy (PE): At the highest points of the swing, the pendulum has maximum gravitational potential energy and zero kinetic energy.

• Kinetic Energy (KE): At the lowest point (equilibrium), the pendulum has maximum kinetic energy and minimum potential energy.

• Graphical Representation: Both PE and KE vary sinusoidally with time, but are out of phase.

• Formula for Gravitational Potential Energy: $PE = mgh$ where m is mass, g is acceleration due to gravity, and h is height above the reference point.

Activity: Elastic Potential Energy Calculations

Spring-Mass System: Energy Conservation

When a block attached to a spring is displaced and released, the system demonstrates conservation of mechanical energy. The sum of kinetic and potential energy remains constant.

• Conservation of Energy: $K_i + U_{sp,i} = K_f + U_{sp,f}$ where $K$ is kinetic energy and $U_{sp}$ is elastic potential energy.

• Kinetic Energy Formula: $K = \frac{1}{2}mv^2$ where m is mass and v is velocity.

• Example Problem:

  • A block is pulled a distance $A$ from equilibrium and released. At different positions, calculate the block's speed using energy conservation.

  • At $x = 0$ (equilibrium): $U_{sp} = 0$, $K = \frac{1}{2}mv^2$ $\frac{1}{2}kA^2 = \frac{1}{2}mv^2$ $v = \sqrt{\frac{kA^2}{m}}$

  • At $x = A/2$: $U_{sp} = \frac{1}{2}k(A/2)^2 = \frac{1}{8}kA^2$ $K = \frac{1}{2}mv^2$ $\frac{1}{2}kA^2 = \frac{1}{8}kA^2 + \frac{1}{2}mv^2$ $v = \sqrt{\frac{3kA^2}{4m}}$

Table: Energy Values at Different Positions

Position (x)	Elastic Potential Energy ($U_{sp}$)	Kinetic Energy ($K$)	Speed ($v$)
$x = A$	$\frac{1}{2}kA^2$	$0$	$0$
$x = 0$	$0$	$\frac{1}{2}kA^2$	$\sqrt{\frac{kA^2}{m}}$
$x = A/2$	$\frac{1}{8}kA^2$	$\frac{3}{8}kA^2$	$\sqrt{\frac{3kA^2}{4m}}$

Additional info: The above table summarizes how energy is distributed between kinetic and potential forms at different positions during oscillation.

Spring Constant and Energy Storage

• Spring Constant (k): Determines the stiffness of the spring. Higher k means more force required for the same displacement.

• Energy Storage: The energy stored in the spring increases with both the spring constant and the square of the displacement.

Summary

• Elastic potential energy in springs and gravitational potential energy in pendulums are key concepts in oscillatory motion.

• Energy conservation allows calculation of speed and energy at any point in the motion.

• Graphs and tables help visualize the periodic exchange between kinetic and potential energy.