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Chapter 10: Problem 48

A body is suspended vertically from an ideal spring of spring constant $2.5 \mathrm{N} / \mathrm{m}$. The spring is initially in its relaxed position. The body is then released and oscillates about its equilibrium position. The motion is described by $$y=(4.0 \mathrm{cm}) \sin [(0.70 \mathrm{rad} / \mathrm{s}) t]$$ What is the maximum kinetic energy of the body?

Short Answer

Expert verified
Answer: The maximum kinetic energy of the body is \(0.050\mathrm{J}\).

Step by step solution

01

Find the amplitude and angular frequency

In the equation of motion, the amplitude (A) and angular frequency (ω) can be found directly: $$y=(4.0\mathrm{cm})\sin[(0.70 \mathrm{rad}/\mathrm{s})t]$$. From this equation, we can see that the amplitude is \(A=4.0 \mathrm{cm}=0.040\mathrm{m}\) and the angular frequency is \(\omega=0.70 \mathrm{rad/s}\).
02

Find the mass of the body

We can find the mass of the body by using the relationship between the spring constant (k), the angular frequency (ω), and the mass (m): $$\omega=\sqrt{\frac{k}{m}}$$. We are given the spring constant \(k=2.5 \mathrm{N/m}\). To find the mass, rearrange the equation and solve for m: $$m=\frac{k}{\omega^2}$$. Now, plug in the values for k and ω: $$m=\frac{2.5\mathrm{N/m}}{(0.70\mathrm{rad/s})^2}$$.
03

Calculate the mass

Calculate the mass using the values provided: $$m=\frac{2.5\mathrm{N/m}}{(0.70\mathrm{rad/s})^2}=5.10\mathrm{kg}$$
04

Calculate the maximum kinetic energy

Now that we have the mass, amplitude, and angular frequency, we can find the maximum kinetic energy (K_max) using the equation for the maximum kinetic energy of a simple harmonic oscillator: $$K_\text{max}=\frac{1}{2}m\omega^2A^2$$. Plug in the values for m, ω, and A: $$K_\text{max}=\frac{1}{2}(5.10\mathrm{kg})(0.70\mathrm{rad/s})^2(0.040\mathrm{m})^2$$.
05

Find the maximum kinetic energy

Calculate the maximum kinetic energy using the values provided: $$K_\text{max}=\frac{1}{2}(5.10\mathrm{kg})(0.70\mathrm{rad/s})^2(0.040\mathrm{m})^2=0.050\mathrm{J}$$. The maximum kinetic energy of the body is \(0.050\mathrm{J}\).

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Most popular questions from this chapter

Equipment to be used in airplanes or spacecraft is often subjected to a shake test to be sure it can withstand the vibrations that may be encountered during flight. A radio receiver of mass \(5.24 \mathrm{kg}\) is set on a platform that vibrates in SHM at \(120 \mathrm{Hz}\) and with a maximum acceleration of $98 \mathrm{m} / \mathrm{s}^{2}(=10 \mathrm{g}) .$ Find the radio's (a) maximum displacement, (b) maximum speed, and (c) the maximum net force exerted on it.
Two steel plates are fastened together using four bolts. The bolts each have a shear modulus of \(8.0 \times 10^{10} \mathrm{Pa}\) and a shear strength of $6.0 \times 10^{8} \mathrm{Pa} .\( The radius of each bolt is \)1.0 \mathrm{cm} .$ Normally, the bolts clamp the two plates together and the frictional forces between the plates keep them from sliding. If the bolts are loose, then the frictional forces are small and the bolts themselves would be subject to a large shear stress. What is the maximum shearing force \(F\) on the plates that the four bolts can withstand? (IMAGE NOT COPY)
The ratio of the tensile (or compressive) strength to the density of a material is a measure of how strong the material is "pound for pound." (a) Compare tendon (tensile strength \(80.0 \mathrm{MPa}\), density $1100 \mathrm{kg} / \mathrm{m}^{3}\( ) with steel (tensile strength \)\left.0.50 \mathrm{GPa}, \text { density } 7700 \mathrm{kg} / \mathrm{m}^{3}\right)$ which is stronger "pound for pound" under tension? (b) Compare bone (compressive strength \(160 \mathrm{MPa}\), density $1600 \mathrm{kg} / \mathrm{m}^{3}$ ) with concrete (compressive strength $\left.0.40 \mathrm{GPa}, \text { density } 2700 \mathrm{kg} / \mathrm{m}^{3}\right):$ which is stronger "pound for pound" under compression?
(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 $\sin ^{2} \omega t+\cos ^{2} \omega t=1.1$.]
A bob of mass \(m\) is suspended from a string of length \(L\) forming a pendulum. The period of this pendulum is \(2.0 \mathrm{s}\) If the pendulum bob is replaced with one of mass \(\frac{1}{3} m\) and the length of the pendulum is increased to \(2 L\), what is the period of oscillation?
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