Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

An object is undergoing SHM with period 1.200 s and amplitude 0.600 m. At \(t =\) 0 the object is at \(x =\) 0 and is moving in the negative \(x\)-direction. How far is the object from the equilibrium position when \(t =\) 0.480 s?

Short Answer

Expert verified
The object is approximately 0.571 m from the equilibrium position at t = 0.480 s.

Step by step solution

01

Formula for SHM Displacement

The equation for the displacement of an object in simple harmonic motion (SHM) is given by:\[ x(t) = A \cos(\omega t + \phi) \]where:- \( A \) is the amplitude.- \( \omega \) is the angular frequency.- \( \phi \) is the phase constant.Given that \( A = 0.600 \) m and the period \( T = 1.200 \) s, we need to find \( \omega \).
02

Calculate Angular Frequency

The angular frequency \( \omega \) can be calculated using the formula:\[ \omega = \frac{2\pi}{T} \]Substitute \( T = 1.200 \) s:\[ \omega = \frac{2\pi}{1.200} = \frac{5\pi}{3} \text{ rad/s} \]
03

Determine Phase Constant φ

At \( t = 0 \), the object is at \( x = 0 \) and moving in the negative x-direction, which suggests a sine function is more suitable for this condition. Therefore, we use:\[ x(t) = A \sin(\omega t) \]However, initially using cosine, for consistency, we have to use:\[ x(0) = A \cos(\phi) = 0 \]which implies \( \phi = \frac{\pi}{2} \) since \( \cos(\frac{\pi}{2}) = 0 \). Adjusting the equation indicates the object is indeed moving in the negative direction.
04

Apply SHM Equation at t = 0.480 s

Substitute \( t = 0.480 \) s and the values \( A = 0.600 \), \( \omega = \frac{5\pi}{3} \), \( \phi = \frac{\pi}{2} \) into the displacement equation:\[ x(0.480) = 0.600 \cos\left(\frac{5\pi}{3} \times 0.480 + \frac{\pi}{2}\right) \]
05

Simplify and Calculate

Calculate the inside of the cosine:\[ \frac{5\pi}{3} \times 0.480 = 0.800\pi \]Then add the phase constant:\[ 0.800\pi + \frac{\pi}{2} = 0.800\pi + 0.500\pi = 1.300\pi \]Now evaluate the cosine:\[ x(0.480) = 0.600 \cos(1.300\pi) = 0.600 \cos(\pi + 0.300\pi) = 0.600 (-\cos(0.300\pi)) \]Where \( \cos(0.300\pi) \approx 0.9511 \), so,\[ x(0.480) = -0.600 \times 0.9511 \approx -0.5707 \text{ m} \]
06

Find Distance from Equilibrium

The distance from the equilibrium position is the absolute value of the displacement:\[ |x(0.480)| = |-0.5707| \approx 0.5707 \text{ m} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
In simple harmonic motion (SHM), the amplitude represents the maximum displacement of the object from its equilibrium position. In essence, it illustrates how far the object moves on either side of this central point.
In our exercise, the amplitude is given as 0.600 m. This means the object can move up to 0.600 meters away from the equilibrium before changing directions.
  • The greater the amplitude, the larger the displacement of the object from equilibrium.
  • Amplitude is crucial as it influences the energy of the system. A higher amplitude means more energy.
When solving problems involving SHM, always remember that while amplitude affects the total energy, it does not influence the period or frequency of the motion.
Angular Frequency
Angular frequency (\(\omega\)) in SHM is a measure of how quickly the object moves through its cycle. It is related to the oscillation period and is expressed in radians per second.
To find angular frequency, use the equation:\[\omega = \frac{2\pi}{T}\]where \(T\) is the period of oscillation. In this exercise, with \(T = 1.200\) s, the angular frequency calculates to:\[\omega = \frac{2\pi}{1.200} = \frac{5\pi}{3}\, \text{rad/s}\]Ul bullet points can be really helpful here!
  • Angular frequency helps determine the speed of oscillations.
  • It's independent of the amplitude but crucial for understanding how cycles progress over time.
Thus, learn always to calculate the angular frequency to comprehend better how fast the system oscillates.
Phase Constant
The phase constant (\(\phi\)) in the context of SHM determines the starting position of the oscillating object relative to its equilibrium. It effectively shifts the wave horizontally on a graph.
Our exercise provides insights into why determining the phase constant is essential. Since the object starts at \(x = 0\) and moves negatively, we use the cosine form,\[A \cos(\phi) = 0\]resulting in \(\phi = \frac{\pi}{2}\).Switching functions or solving for \(\phi\) depends on initial conditions:
  • The phase constant is often chosen to match initial displacement and velocity conditions.
  • A correct phase constant aligns theoretical models to actual movements.
Understanding phase constants allow for more accurate predictions of position over time.
Cosine Function
In SHM, the cosine function is often used to model the displacement of an object over time. The function:\[x(t) = A \cos(\omega t + \phi)\]captures how position changes with time using amplitude, angular frequency, and phase constant.
In this exercise, translating theoretical equations to practical values involved:
  • Substituting given quantities such as amplitude, angular frequency, and phase constant.
  • Evaluating the equation at specific times to find its current position.
Ultimately, cosine functions are pivotal for calculating how far an object is from equilibrium at any given moment, making it central to analyzing SHM.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

You hang various masses \(m\) from the end of a vertical, 0.250-kg spring that obeys Hooke's law and is tapered, which means the diameter changes along the length of the spring. Since the mass of the spring is not negligible, you must replace \(m\) in the equation \(T =\) 2\(\pi\sqrt{ m/k }\) with \(m + m_\mathrm{eff}\), where \(m_\mathrm{eff}\) is the effective mass of the oscillating spring. (See Challenge Problem 14.93.) You vary the mass m and measure the time for 10 complete oscillations, obtaining these data: (a) Graph the square of the period \(T\) versus the mass suspended from the spring, and find the straight line of best fit. (b) From the slope of that line, determine the force constant of the spring. (c) From the vertical intercept of the line, determine the spring's effective mass. (d) What fraction is \(m_\mathrm{eff}\) of the spring's mass? (e) If a 0.450-kg mass oscillates on the end of the spring, find its period, frequency, and angular frequency.

A 50.0-g hard-boiled egg moves on the end of a spring with force constant \(k =\) 25.0 N/m. Its initial displacement is 0.300 m. A damping force \(F_x = -bv_x\) acts on the egg, and the amplitude of the motion decreases to 0.100 m in 5.00 s. Calculate the magnitude of the damping constant \(b\).

A 1.50-kg mass on a spring has displacement as a function of time given by $$x(t) = 7.40 \mathrm{cm}) \mathrm{cos} [ (4.16 \mathrm{rad}/s)t - 2.42] $$ Find (a) the time for one complete vibration; (b) the force constant of the spring; (c) the maximum speed of the mass; (d) the maximum force on the mass; (e) the position, speed, and acceleration of the mass at \(t =\) 1.00 s; (f) the force on the mass at that time.

A 2.00-kg bucket containing 10.0 kg of water is hanging from a vertical ideal spring of force constant 450 N/m and oscillating up and down with an amplitude of 3.00 cm. Suddenly the bucket springs a leak in the bottom such that water drops out at a steady rate of 2.00 g/s. When the bucket is half full, find (a) the period of oscillation and (b) the rate at which the period is changing with respect to time. Is the period getting longer or shorter? (c) What is the shortest period this system can have?

A spring of negligible mass and force constant \(k =\) 400 N/m is hung vertically, and a 0.200-kg pan is suspended from its lower end. A butcher drops a 2.2-kg steak onto the pan from a height of 0.40 m. The steak makes a totally inelastic collision with the pan and sets the system into vertical SHM. What are (a) the speed of the pan and steak immediately after the collision; (b) the amplitude of the subsequent motion; (c) the period of that motion?

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free