# PHYS 1P21 Lecture 46: Elastic potential energy and Damped oscillations

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Simple Harmonic Motion - Elastic potential energy and Damped oscillations

Elastic potential energy

An argument similar in spirit to the one above for gravitational potential energy shows that the

formula for elastic potential energy is (see page 285ff in Section 10.3 of the textbook for details;

calculus lovers, antidifferentiate Hooke's law)

where xis the displacement of the spring from its equilibrium position

and kis the stiffness constant of the spring.

Where is potential energy stored? Answering this question leads to the concept of a force field;

we'll discuss force fields in PHYS

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Exercises

8. Consider a block of mass 5.1 kg attached to a spring. The position-time function of this

oscillator is x= 8.2 sin(2.7t), where xis measured in cm and tis measured in seconds.

(a) Determine the total mechanical energy of the oscillator.

(b) Determine the stiffness constant of the spring.

(c) Determine the maximum potential energy.

(d) Determine the maximum kinetic energy.

(e) Determine the positions at which the kinetic energy and the potential energy of the oscillator

are equal.

Pendulum motion

Recall from earlier in these lecture notes that for a block on the end of a spring, applying

Newton's law to the block results in

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This is an example of a differential equation, and to solve a differential equation means to

determine a position function x(t) that satisfies the equation. You'll learn how to do this in

second-year physics (PHYS 2P20), and second-year math (MATH 2P08), but for now you can

verify that position functions of the form

and

both satisfy the differential equation above, provided that satisfies a certain condition (the same

condition that was observed earlier in the notes).

From a different perspective, we can also infer that if a physical phenomenon is modelled by a

differential equation of the form given above, the phenomenon is an example of simple harmonic

motion.

Let's consider a simple pendulum. First draw a free-body diagram:

In the radial direction, applying Newton's second law gives:

In the tangential direction, applying Newton's second law gives:

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