Chapter 13: Problem 11
Why is critical damping desirable in a car's suspension?
Chapter 13: Problem 11
Why is critical damping desirable in a car's suspension?
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Get started for freeThe total energy of a mass-spring system is the sum of its kinetic and potential energy: \(E=\frac{1}{2} m v^{2}+\frac{1}{2} k x^{2} .\) Assuming \(E\) remains constant, differentiate both sides of this expression with respect to time and show that Equation 13.3 results. (Hint: Remember that \(v=d x / d t .)\)
An object undergoes simple harmonic motion in two mutually perpendicular directions, its position given by \(\vec{r}=A \sin \omega t \hat{\imath}+\) A cos wit. (a) Show that the object remains a fixed distance from the origin (i.e.. that its path is circular), and find that distance. (b) Find an expression for the object's velocity. (c) Show that the speed remains constant, and find its value. (d) Find the angular speed of the object in its circular path.
At the heart of a grandfather clock is a simple pendulum \(1.45 \mathrm{m}\) long; the clock ticks each time the pendulum reaches its maximum displacement in either direction. What's the time interval between ticks?
You're working on the script of a movie whose plot involves a hole drilled straight through Earth's center and out the other side. You're asked to determine what will happen if a person falls into the hole. You find that the gravitational acceleration inside Earth points toward Earth's center, with magnitude given approximately by \(g(r)=g_{0}\left(r / R_{\mathrm{E}}\right),\) where \(g_{0}\) is the surface value, \(r\) is the distance from Earth's center, and \(R_{\mathrm{E}}\) is Earth's radius. What do you report for the person's motion, including equations and values for any relevant parameters?
What happens to the frequency of a simple harmonic oscillator when the spring constant is doubled? When the mass is doubled?
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