Problem 28 - University Physics with Modern Physics

Question

A cheerleader waves her pom-pom in SHM with an amplitude of 18.0 \(\mathrm{cm}\) and a frequency of 0.850 \(\mathrm{Hz}\) . Find (a) the maximum magnitude of the acceleration and of the velocity; (b) the acceleration and speed when the pom- pom's coordinate is \(x=+9.0 \mathrm{cm} ;(\mathrm{c})\) the time required to move from the equilibrium position directly to a point 12.0 \(\mathrm{cm}\) away. (d) Which of the quantities asked for in parts (a), (b), and (c) can be found using the energy approach used in Section \(14.3,\) and which cannot? Explain.

Step-by-Step Solution

Answer

Max velocity: 96.1 cm/s; Max acceleration: 512 cm/s²; Speed at x=9.0 cm: 83.2 cm/s; Time to 12.0 cm: 0.085s. Energy approach finds maximum velocity and speeds at positions.

Step 1: Understanding Simple Harmonic Motion (SHM) Parameters

In SHM, the position of the object as a function of time is given by \( x(t) = A\cos(2\pi ft + \phi) \). Here \(A = 18.0 \text{ cm} \), \(f = 0.850 \text{ Hz} \). We can find \(\omega = 2\pi f\), which is the angular frequency.

Step 2: Finding Maximum Velocity and Acceleration

Maximum velocity \(v_{max}\) occurs when \(x = 0\) and is given by \( v_{max} = A\omega \). Maximum acceleration \(a_{max}\) is when the displacement is maximum, given by \( a_{max} = A\omega^2 \). Calculate \( \omega = 2\pi \times 0.850 \text{ Hz} \) to find these values.

Step 3: Calculating Acceleration and Speed at x = +9.0 cm

At \( x = 9.0 \text{ cm} \), speed \( v \) can be found using \( v^2 = \omega^2 (A^2 - x^2) \) and acceleration \( a \) is \( a = -\omega^2 x \). Substitute \( x = 9.0 \text{ cm} \) to find these values.

Step 4: Calculating Time to Move 12.0 cm from Equilibrium

To find the time needed to move from 0 to 12.0 cm, use the SHM equation \( x(t) = A\cos(\omega t) \), solve for \( t \) by substituting \( x = 12.0 \text{ cm} \) and rearranging to get \( \cos^{-1}(x/A) = \omega t \). Calculate \( t \) by solving this.

Step 5: Determine Which Quantities Can Be Found Using Energy Approach

The energy approach, based on the idea that \(\frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2\), can be used to find maximum velocity, speed at a position, and acceleration indirectly via energy equations. However, time to reach a specific point is not directly solved using energy and requires the equation of motion.

Key Concepts

Maximum Velocity

In Simple Harmonic Motion (SHM), the maximum velocity of an oscillating object occurs as it passes through its equilibrium position, where the displacement is zero. Imagine a cheerleader waving a pom-pom back and forth. As the pom-pom swings through the midpoint of its path, it reaches its highest speed.

The formula for maximum velocity in SHM is given by:

• \( v_{max} = A\omega \)

Where:

• \( A \) is the amplitude of the motion
• \( \omega \) is the angular frequency

To find the angular frequency \( \omega \), use the relation \( \omega = 2\pi f \), where \( f \) is the frequency. Thus, if the amplitude \( A \) is 18 cm and the frequency \( f \) is 0.850 Hz, you can calculate \( \omega \) and consequently the maximum velocity. This concept is key in determining how fast an object in SHM can move.

Maximum Acceleration

Just like velocity, acceleration in SHM also reaches its maximum but when the displacement is also at its maximum, rather than at equilibrium. As the pom-pom reaches the extremities of its path, it experiences its greatest acceleration because it's about to change direction.

The formula for maximum acceleration in SHM is:

• \( a_{max} = A\omega^2 \)

This is derived from the fact that the restoring force, and hence the acceleration, is greatest when the displacement is largest. By knowing the amplitude \( A \) and angular frequency \( \omega \), you can calculate the maximum acceleration experienced by the object. It's fascinating how in SHM, both velocity and acceleration are closely tied to the amplitude and angular frequency.

SHM Equations

The behaviors of objects in Simple Harmonic Motion can be precisely described using mathematical equations. These equations enable us to predict how the motion evolves over time. An important equation in SHM is the expression for displacement:

• \( x(t) = A\cos(2\pi ft + \phi) \)

In this equation:

• \( x(t) \) is the object's position at time \( t \)
• \( A \) is the amplitude
• \( f \) is the frequency
• \( \phi \) is the phase constant, which depends on initial conditions

Additionally, we use equations like:

• \( v(t) = -A\omega\sin(\omega t + \phi) \)
• \( a(t) = -A\omega^2\cos(\omega t + \phi) \)

These describe the velocity and acceleration at any given time \( t \). Mastering these equations is essential for understanding and predicting the behavior of oscillatory systems.

Energy Approach

In the analysis of Simple Harmonic Motion, the energy approach is incredibly useful. It is based on the principle of conservation of energy. At any point during the oscillation, the total mechanical energy is constant. The energy is the sum of kinetic and potential energy, given by:

• \( \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2 \)

Where:

• \( m \) is the mass of the object
• \( v \) is the velocity
• \( k \) is the spring constant
• \( x \) is the displacement
• \( A \) is the amplitude

This approach helps us determine properties such as maximum velocity and speed at specific positions without directly using time or displacement-based equations. However, it does not help directly in determining the time taken to reach a specific displacement, which necessitates using the SHM motion equations directly.

Angular Frequency

Angular frequency, denoted \( \omega \), is a fundamental concept in analyzing Simple Harmonic Motion. It connects to how frequently the oscillation cycles through its path in radian measure per second. Unlike regular frequency, which tells us how many cycles occur per second, angular frequency describes the rate of rotation within the cycle itself.

The formula for angular frequency is:

• \( \omega = 2\pi f \)

In this equation:

• \( f \) is the frequency of the oscillation

Angular frequency is crucial because it directly influences both maximum velocity and acceleration in SHM, as seen in the equations \( v_{max} = A\omega \) and \( a_{max} = A\omega^2 \). Understanding \( \omega \) gives deep insight into the dynamics of oscillating systems.