Problem 65 - Calculus Early Transcendentals

Question

If an object suspended from a spring is displaced vertically from its equilibrium position by a small amount and released, and if the air resistance and the mass of the spring are ignored, then the resulting oscillation of the object is called simple harmonic motion. Under appropriate conditions the displacement \(y\) from equilibrium in terms of time \(t\) is given by $$ y=A \cos \omega t $$ where \(A\) is the initial displacement at time \(t=0,\) and \(\omega\) is a constant that depends on the mass of the object and the stiffness of the spring (see the accompanying figure). The constant \(|A|\) is called the amplitude of the motion and \(\omega\) the angular frequency. (a) Show that $$ \frac{d^{2} y}{d t^{2}}=-\omega^{2} y $$ (b) The period \(T\) is the time required to make one complete oscillation. Show that \(T=2 \pi / \omega\). (c) The frequency \(f\) of the vibration is the number of oscillations per unit time. Find \(f\) in terms of the period \(T\) (d) Find the amplitude, period, and frequency of an object that is executing simple harmonic motion given by \(y=0.6 \cos 15 t,\) where \(t\) is in seconds and \(y\) is in centimeters.

Step-by-Step Solution

Answer

Verified differential equation, period is \( T = \frac{2\pi}{\omega} \), frequency is \( f = \frac{\omega}{2\pi} \), given motion values: amplitude 0.6 cm, period \( \frac{2\pi}{15} \) sec, frequency \( \frac{15}{2\pi} \) Hz.

Step 1: Verify the differential equation

To show that \( \frac{d^2 y}{dt^2} = -\omega^2 y \), we first need to compute the first and second derivatives of \( y = A \cos(\omega t) \). Start by computing the first derivative: \( \frac{dy}{dt} = -A \omega \sin(\omega t) \). Then, the second derivative is \( \frac{d^2 y}{dt^2} = -A \omega^2 \cos(\omega t) \). Since \( y = A \cos(\omega t) \), we can substitute back to get \( \frac{d^2 y}{dt^2} = -\omega^2 y \). Thus, the differential equation is verified.

Step 2: Derive the period formula

To find the period \( T \), recall the definition that one full oscillation corresponds to a full cycle of the cosine function, from 0 to \( 2\pi \). The argument \( \omega t \) must therefore change by \( 2\pi \), i.e., \( \omega T = 2\pi \). Solving this equation for \( T \), we get \[ T = \frac{2\pi}{\omega} \].

Step 3: Relate frequency to period

The frequency \( f \) is the inverse of the period \( T \). Thus, \( f = \frac{1}{T} \). Substituting the expression found for \( T \), we get \( f = \frac{\omega}{2\pi} \).

Step 4: Identify parameters for given motion

For the given motion \( y = 0.6 \cos(15t) \), compare it with \( y = A \cos(\omega t) \) to identify that \( A = 0.6 \) cm and \( \omega = 15 \) rad/sec. The amplitude is \( 0.6 \) cm. The period \( T = \frac{2\pi}{\omega} = \frac{2\pi}{15} \) seconds. The frequency \( f = \frac{1}{T} = \frac{15}{2\pi} \) Hz.

Key Concepts

Differential Equations

Differential equations are mathematical expressions involving derivatives, which describe how a quantity changes with respect to another. In simple harmonic motion, we encounter a second-order differential equation that characterizes how displacement changes over time. Here, the equation \[\frac{d^{2} y}{d t^{2}} = -\omega^{2} y\]relates the acceleration of the object to its displacement. To break it down:

• The left side, \( \frac{d^{2} y}{d t^{2}} \), represents acceleration, which is the second derivative of displacement \( y \).
• The right side, \(-\omega^{2} y \), involves the angular frequency squared and the negative sign indicates a restoring force towards equilibrium.

Understanding this equation helps us see how acceleration is always directed opposite to displacement, maintaining a harmonic oscillation when no other forces (like air resistance) are acting. By solving this differential equation, we find how the system behaves over time.

Angular Frequency

Angular frequency \( \omega \) plays a crucial role in analyzing simple harmonic motion. It describes how fast the object oscillates in radians per second. Given by the formula \[ \omega = 2 \pi f\]it links the concept of radians and traditional frequency (cycles per second) together.

• Unlike regular frequency, which counts full oscillations per second, angular frequency expresses the "how fast" in radians per second.
• The greater the angular frequency, the faster the system oscillates.

Knowing \( \omega \) allows us to predict the motion over time, as it helps determine the speed of oscillation, thereby linking it to the dynamics described by the differential equation in the previous section.

Amplitude

Amplitude in simple harmonic motion measures the maximum displacement from the equilibrium position. In the context of our situation, amplitude \( A \) is the peak height or depth reached during oscillations.

• This value indicates the energy stored in the motion; higher amplitude means more energy.
• Specifically for formulas like \( y = A \cos\omega t \), amplitude is the absolute value \( |A| \).

Understanding amplitude provides insight into the scale of motion and stability of the system. In practical terms, knowing the amplitude helps determine safety margins for mechanical parts exposed to oscillating forces.

Oscillation Period

The period \( T \) of an oscillation is the time it takes for a complete cycle of motion to return to the same position and state. It can be visualized as the time taken for one full trip along the path of movement. The formula for period derived from angular frequency is \[ T = \frac{2 \pi}{\omega}.\]

• This shows the direct relationship between period and angular frequency: higher \( \omega \) results in a shorter period.
• Period is a fundamental characteristic that helps compare how quickly different systems oscillate.

Knowing the period helps predict when the next cycle will begin, which is particularly useful for designing and analyzing cyclic systems, such as clock pendulums or tuned car suspensions.

Frequency

Frequency \( f \) describes the number of oscillations that occur within a unit of time, usually seconds. It's calculated as the inverse of the period, given by \[ f = \frac{1}{T}.\]

• Measured in hertz (Hz), where 1 Hz equals 1 cycle per second.
• Higher frequency means more oscillations occur in the same period.

In engineering, frequency is crucial since many systems are designed to work optimally at certain frequencies. It's also used in signal processing, communications, and any application involving cyclic behaviors, showing its wide-ranging importance in technology and natural phenomena.