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You are watching an object that is moving in SHM. When the object is displaced 0.600 m to the right of its equilibrium position, it has a velocity of 2.20 m/s to the right and an acceleration of 8.40 m/s\(^2\) to the left. How much farther from this point will the object move before it stops momentarily and then starts to move back to the left?

Short Answer

Expert verified
0.240 m

Step by step solution

01

Identify Given Values

We start by identifying the given details of the problem. The object is in Simple Harmonic Motion (SHM):- Displacement from equilibrium, \( x = 0.600 \ \text{m} \) (to the right),- Velocity, \( v = 2.20 \ \text{m/s} \) (to the right),- Acceleration, \( a = -8.40 \ \text{m/s}^2 \) (to the left).
02

Use Acceleration Formula in SHM

In SHM, acceleration is given by \( a = - \omega^2 x \). Rearranging to solve for angular frequency \( \omega \), we have:\[a = -\omega^2 x \8.40 = \omega^2 \times 0.600 \\omega^2 = \frac{8.40}{0.600} = 14 \\omega = \sqrt{14}\]
03

Calculate Maximum Displacement

In SHM, when the object stops momentarily, it is at its maximum displacement \( x_{max} \). Using energy conservation, total energy at maximum displacement is equal to total energy at \( x = 0.600 \ \text{m} \):\[\frac{1}{2}m\omega^2 x_{max}^2 = \frac{1}{2}m\omega^2 x^2 + \frac{1}{2}mv^2 \\omega^2 x_{max}^2 = \omega^2 x^2 + v^2 \x_{max}^2 = x^2 + \frac{v^2}{\omega^2} \x_{max}^2 = (0.600)^2 + \frac{(2.20)^2}{14} \x_{max}^2 = 0.360 + 0.346 \x_{max} = \sqrt{0.706} \]
04

Solve for Additional Displacement

Now calculate the additional distance from the given point:\[ \text{Maximum displacement: } x_{max} = \sqrt{0.706} \approx 0.840 \ \text{m} \\text{Initial displacement: } x = 0.600 \ \text{m} \\text{Additional distance = } x_{max} - x = 0.840 - 0.600 \]
05

Final Calculation

Perform the subtraction to find the additional displacement:\[\text{Additional displacement } = 0.240 \ \text{m}\]

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Key Concepts

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

Angular Frequency
In Simple Harmonic Motion (SHM), angular frequency is a crucial concept. It is represented by the symbol \( \omega \) and serves as a measure of how quickly a system oscillates. Angular frequency can be derived from the formula \( a = - \omega^2 x \), where \( a \) is the acceleration at displacement \( x \). This formula shows the relationship of acceleration to angular frequency and displacement.For example, in our problem, at a displacement of 0.600 meters, the object has an acceleration of -8.40 m/s\(^2\). Solving \( 8.40 = \omega^2 \times 0.600 \) gives \( \omega^2 = 14 \), leading to \( \omega = \sqrt{14} \). Understanding this concept allows us to predict how fast the oscillations happen and to link this to energy and displacement in SHM.
Maximum Displacement
Maximum displacement in SHM refers to the furthest point the object reaches from its equilibrium position before momentarily coming to rest. This point marks the peak of its oscillation where velocity is zero.We use energy conservation principles to calculate this. Both potential and kinetic energies are in play, but at maximum displacement, all energy is potential. The formula to find maximum displacement from the point at \( x = 0.600 \, \text{m} \) is given by:\[ \omega^2 x_{max}^2 = \omega^2 x^2 + v^2 \] Solving gives \[ x_{max}^2 = 0.360 + 0.346 \quad \Rightarrow \quad x_{max} = \sqrt{0.706} \approx 0.840 \, \text{m} \] This shows the farthest point the object reaches before reversing direction.
Energy Conservation
Energy conservation in SHM is essential to understanding how the system oscillates. The total mechanical energy remains constant; it simply shifts between kinetic and potential forms.At any point, the energy \( E \) is\[ E = \frac{1}{2} m \omega^2 x^2 + \frac{1}{2} mv^2 \] At maximum displacement, kinetic energy is zero, and all energy is potential:\[ E = \frac{1}{2} m \omega^2 x_{max}^2 \] The conservation principle lets us equate these energies, allowing us to solve for variables like \( x_{max} \). This principle helps explain how the changing energy forms maintain the harmonic motion.
Velocity
Velocity plays a dynamic role in SHM as it varies continuously with displacement. While displacement and acceleration at any point can determine velocity, in SHM, these components oscillate in sinusoidal patterns.From its maximum when passing through equilibrium to zero at maximum displacement, velocity is crucial for analyzing motion. Given:\[ v = 2.20 \, \text{m/s at} \, x = 0.600 \, \text{m} \] This velocity helps us set the energy equations by providing kinetic energy, which partners with potential energy to maintain the motion. Observing these oscillations reveals insights into the system's dynamics. Understanding this shows how all components interrelate in harmonic motion.

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Most popular questions from this chapter

A 5.00-kg partridge is suspended from a pear tree by an ideal spring of negligible mass. When the partridge is pulled down 0.100 m below its equilibrium position and released, it vibrates with a period of 4.20 s. (a) What is its speed as it passes through the equilibrium position? (b) What is its acceleration when it is 0.050 m above the equilibrium position? (c) When it is moving upward, how much time is required for it to move from a point 0.050 m below its equilibrium position to a point 0.050 m above it? (d) The motion of the partridge is stopped, and then it is removed from the spring. How much does the spring shorten?

A large, 34.0-kg bell is hung from a wooden beam so it can swing back and forth with negligible friction. The bell's center of mass is 0.60 m below the pivot. The bell's moment of inertia about an axis at the pivot is 18.0 kg \(\cdot\) m\(^2\). The clapper is a small, 1.8-kg mass attached to one end of a slender rod of length \(L\) and negligible mass. The other end of the rod is attached to the inside of the bell; the rod can swing freely about the same axis as the bell. What should be the length \(L\) of the clapper rod for the bell to ring silently\(-\)that is, for the period of oscillation for the bell to equal that of the clapper?

An object with height \(h\), mass \(M\), and a uniform cross-sectional area \(A\) floats upright in a liquid with density \(\rho\). (a) Calculate the vertical distance from the surface of the liquid to the bottom of the floating object at equilibrium. (b) A downward force with magnitude \(F\) is applied to the top of the object. At the new equilibrium position, how much farther below the surface of the liquid is the bottom of the object than it was in part (a)? (Assume that some of the object remains above the surface of the liquid.) (c) Your result in part (b) shows that if the force is suddenly removed, the object will oscillate up and down in SHM. Calculate the period of this motion in terms of the density \(\rho\) of the liquid, the mass \(M\), and the cross- sectional area A of the object. You can ignore the damping due to fluid friction (see Section 14.7).

When a 0.750-kg mass oscillates on an ideal spring, the frequency is 1.75 Hz. What will the frequency be if 0.220 kg are (a) added to the original mass and (b) subtracted from the original mass? Try to solve this problem \(without\) finding the force constant of the spring.

While on a visit to Minnesota ("Land of 10,000 Lakes"), you sign up to take an excursion around one of the larger lakes. When you go to the dock where the 1500-kg boat is tied, you find that the boat is bobbing up and down in the waves, executing simple harmonic motion with amplitude 20 cm. The boat takes 3.5 s to make one complete up-and-down cycle. When the boat is at its highest point, its deck is at the same height as the stationary dock. As you watch the boat bob up and down, you (mass 60 kg) begin to feel a bit woozy, due in part to the previous night's dinner of lutefisk. As a result, you refuse to board the boat unless the level of the boat's deck is within 10 cm of the dock level. How much time do you have to board the boat comfortably during each cycle of up-and-down motion?

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