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Chapter 10: Problem 27

The period of oscillation of a spring-and-mass system is \(0.50 \mathrm{s}\) and the amplitude is \(5.0 \mathrm{cm} .\) What is the magnitude of the acceleration at the point of maximum extension of the spring?

Short Answer

Expert verified
Answer: The magnitude of the acceleration at the point of maximum extension of the spring is 25.13 m/s².

Step by step solution

01

Write down the known values

First, let's list down the given values: - Period of oscillation (T) = \(0.50s\) - Amplitude (A) = \(5.0cm\)
02

Calculate the angular frequency (ω)

We can find the angular frequency (ω) from the period using the relation: \(ω = \dfrac{2 \pi}{T}\) Now, plugging in the value of T: \(ω = \dfrac{2 \pi}{0.50s} = 4 \pi\:s^{-1}\)
03

Find the maximum displacement (X_max)

The maximum displacement of the mass occurs at the point of maximum extension, which is also equal to the amplitude A: \(X_{max} = 5.0 cm = 0.05 m\)
04

Calculate the maximum acceleration (a_max)

At the point of maximum extension, the mass experiences maximum acceleration. We can find the maximum acceleration using the general formula for the acceleration in Simple Harmonic Motion: \(a_{max} = -ω^2 X_{max}\) Now plug in the values of \(ω\) and \(X_{max}\): \(a_{max} = -(4\pi)^2 \times 0.05 = -25.13 m/s^2\) The negative sign indicates that the acceleration is opposite to the direction of the maximum displacement.
05

Report the magnitude of the acceleration

Since we only need to find the magnitude of the acceleration, we can ignore the negative sign: Magnitude of the acceleration at the point of maximum extension of the spring = \(25.13 \:m/s^2\)

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Most popular questions from this chapter

Luke is trying to catch a pesky animal that keeps eating vegetables from his garden. He is building a trap and needs to use a spring to close the door to his trap. He has a spring in his garage and he wants to determine the spring constant of the spring. To do this, he hangs the spring from the ceiling and measures that it is \(20.0 \mathrm{cm}\) long. Then he hangs a \(1.10-\mathrm{kg}\) brick on the end of the spring and it stretches to $31.0 \mathrm{cm} .$ (a) What is the spring constant of the spring? (b) Luke now pulls the brick \(5.00 \mathrm{cm}\) from the equilibrium position to watch it oscillate. What is the maximum speed of the brick? (c) When the displacement is \(2.50 \mathrm{cm}\) from the equilibrium position, what is the speed of the brick? (d) How long will it take for the brick to oscillate five times?
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?
What is the period of a pendulum formed by placing a horizontal axis (a) through the end of a meterstick ( 100 -cm mark)? (b) through the 75 -cm mark? (c) through the \(60-\mathrm{cm}\) mark?
A marble column with a cross-sectional area of \(25 \mathrm{cm}^{2}\) supports a load of \(7.0 \times 10^{4} \mathrm{N} .\) The marble has a Young's modulus of \(6.0 \times 10^{10} \mathrm{Pa}\) and a compressive strength of $2.0 \times 10^{8} \mathrm{Pa} .$ (a) What is the stress in the column? (b) What is the strain in the column? (c) If the column is \(2.0 \mathrm{m}\) high, how much is its length changed by supporting the load? (d) What is the maximum weight the column can support?
A baby jumper consists of a cloth seat suspended by an elastic cord from the lintel of an open doorway. The unstretched length of the cord is $1.2 \mathrm{m}\( and the cord stretches by \)0.20 \mathrm{m}$ when a baby of mass \(6.8 \mathrm{kg}\) is placed into the seat. The mother then pulls the seat down by \(8.0 \mathrm{cm}\) and releases it. (a) What is the period of the motion? (b) What is the maximum speed of the baby?
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