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Chapter 10: Problem 32

A \(170-g\) object on a spring oscillates left to right on a frictionless surface with a frequency of \(3.00 \mathrm{Hz}\) and an amplitude of $12.0 \mathrm{cm} .$ (a) What is the spring constant? (b) If the object starts at \(x=12.0 \mathrm{cm}\) at \(t=0\) and the equilibrium point is at \(x=0,\) what equation describes its position as a function of time?

Short Answer

Expert verified
Answer: The spring constant (k) is 53.94 N/m. The equation that describes the object's position as a function of time is x(t) = 0.12*cos(18.85*t).

Step by step solution

01

Convert the mass and amplitude to SI units

As we are going to deal with SI units, we need to convert the mass and amplitude to the respective SI units. Mass: 170 g = 0.17 kg Amplitude: 12.0 cm = 0.12 m
02

Calculate the angular frequency (ω)

The problem provides us with the frequency (f) in Hz, and we can calculate the angular frequency using the formula ω = 2πf. ω = 2π * 3.00 Hz = 18.85 rad/s
03

Calculate the spring constant (k)

Now, we can use the formula ω = sqrt(k/m) to find the spring constant (k). Rearranging the formula to solve for k: k = ω²*m k = (18.85 rad/s)² * 0.17 kg = 53.94 N/m So, the spring constant is 53.94 N/m.
04

Determine the phase constant (φ)

Since the object starts at the maximum amplitude position (x = 12.0 cm = 0.12 m) and moves left at t = 0, the phase constant (φ) should be zero, as the cosine function is at its peak at 0. φ = 0
05

Write the equation for the position as a function of time

Now that we have all the components, we can write the equation for the position of the object as a function of time: x(t) = A*cos(ω*t + φ) x(t) = 0.12*cos(18.85*t) So, the equation that describes the object's position as a function of time is x(t) = 0.12*cos(18.85*t).

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Most popular questions from this chapter

A mass-spring system oscillates so that the position of the mass is described by \(x=-10 \cos (1.57 t),\) where \(x\) is in \(\mathrm{cm}\) when \(t\) is in seconds. Make a plot that has a dot for the position of the mass at $t=0, t=0.2 \mathrm{s}, t=0.4 \mathrm{s}, \ldots, t=4 \mathrm{s}$ The time interval between each dot should be 0.2 s. From your plot, tell where the mass is moving fastest and where slowest. How do you know?
A grandfather clock is constructed so that it has a simple pendulum that swings from one side to the other, a distance of \(20.0 \mathrm{mm},\) in $1.00 \mathrm{s} .$ What is the maximum speed of the pendulum bob? Use two different methods. First, assume SHM and use the relationship between amplitude and maximum speed. Second, use energy conservation.
A \(230.0-\mathrm{g}\) object on a spring oscillates left to right on a frictionless surface with a frequency of \(2.00 \mathrm{Hz}\). Its position as a function of time is given by \(x=(8.00 \mathrm{cm})\) sin \(\omega t\) (a) Sketch a graph of the elastic potential energy as a function of time. (b) The object's velocity is given by \(v_{x}=\omega(8.00 \mathrm{cm}) \cos \omega t .\) Graph the system's kinetic energy as a function of time. (c) Graph the sum of the kinetic energy and the potential energy as a function of time. (d) Describe qualitatively how your answers would change if the surface weren't frictionless.
A sphere of copper is subjected to 100 MPa of pressure. The copper has a bulk modulus of 130 GPa. By what fraction does the volume of the sphere change? By what fraction does the radius of the sphere change?
(a) Given that the position of an object is \(x(t)=A\) cos \(\omega t\) show that \(v_{x}(t)=-\omega A\) sin \(\omega t .\) [Hint: Draw the velocity vector for point \(P\) in Fig. \(10.17 \mathrm{b}\) and then find its \(x\) component.] (b) Verify that the expressions for \(x(t)\) and \(v_{x}(t)\) are consistent with energy conservation. [Hint: Use the trigonometric identity $\sin ^{2} \omega t+\cos ^{2} \omega t=1.1$.]
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