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Chapter 10: Problem 55

What is the period of a pendulum consisting of a \(6.0-\mathrm{kg}\) mass oscillating on a 4.0 -m-long string?

Short Answer

Expert verified
Answer: The period of the pendulum is approximately 4.03 s.

Step by step solution

01

Identify the given values and the formula to use

We are given the mass of the pendulum (\(6.0\, \mathrm{kg}\)) and the length of the string (\(4.0\, \mathrm{m}\)). We need to find the period (T) of the pendulum. The formula to use is: \(T = 2\pi\sqrt{\frac{\mathrm{L}}{\mathrm{g}}}\).
02

Plug in the given values

Now, substitute the given values into the formula: \(T = 2\pi\sqrt{\frac{4.0\, \mathrm{m}}{9.81\, \mathrm{m/s^2}}}\).
03

Calculate the period

Calculate the square root of the ratio and then multiply by \(2\pi\): \(T \approx 2\pi\sqrt{0.408} \approx 4.03\, \mathrm{s}\).
04

Write the answer

The period of the pendulum with a \(6.0\, \mathrm{kg}\) mass oscillating on a \(4.0\, \mathrm{m}\)-long string is approximately \(4.03\, \mathrm{s}\).

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Most popular questions from this chapter

The gravitational potential energy of a pendulum is \(U=m g y .\) (a) Taking \(y=0\) at the lowest point, show that \(y=L(1-\cos \theta),\) where \(\theta\) is the angle the string makes with the vertical. (b) If \(\theta\) is small, \((1-\cos \theta)=\frac{1}{2} \theta^{2}\) and \(\theta=x / L\) (Appendix A.7). Show that the potential energy can be written \(U=\frac{1}{2} k x^{2}\) and find the value of \(k\) (the equivalent of the spring constant for the pendulum).
Spider silk has a Young's modulus of $4.0 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}\( and can withstand stresses up to \)1.4 \times 10^{9} \mathrm{N} / \mathrm{m}^{2} . \mathrm{A}$ single webstrand has across-sectional area of \(1.0 \times 10^{-11} \mathrm{m}^{2}\) and a web is made up of 50 radial strands. A bug lands in the center of a horizontal web so that the web stretches downward. (a) If the maximum stress is exerted on each strand, what angle \(\theta\) does the web make with the horizontal? (b) What does the mass of a bug have to be in order to exert this maximum stress on the web? (c) If the web is \(0.10 \mathrm{m}\) in radius, how far down does the web extend? (IMAGE NOT COPY)
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?
The diaphragm of a speaker has a mass of \(50.0 \mathrm{g}\) and responds to a signal of frequency \(2.0 \mathrm{kHz}\) by moving back and forth with an amplitude of \(1.8 \times 10^{-4} \mathrm{m}\) at that frequency. (a) What is the maximum force acting on the diaphragm? (b) What is the mechanical energy of the diaphragm?
An anchor, made of cast iron of bulk modulus \(60.0 \times 10^{9} \mathrm{Pa}\) and of volume \(0.230 \mathrm{m}^{3},\) is lowered over the side of the ship to the bottom of the harbor where the pressure is greater than sea level pressure by \(1.75 \times 10^{6} \mathrm{Pa} .\) Find the change in the volume of the anchor.
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