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Chapter 8: Problem 3

The angle turned through by the flywheel of a generator during a time interval \(t\) is given by $$ \phi=a t+b t^{3}-c t^{4}, $$ where \(a, b\), and \(c\) are constants. What is the expression for its (a) angular velocity and ( \(b\) ) angular acceleration?

Short Answer

Expert verified
The angular velocity \(w = a + 3bt^{2} - 4ct^{3}\) and the angular acceleration \(\alpha = 6bt - 12ct^{2}\).

Step by step solution

01

Compute the derivative of the angular displacement to find the angular velocity

The angular velocity \(w\) is the derivative of the angular displacement \(\phi\) with respect to time: \(w = \frac{d\phi}{dt}\). Therefore, differentiate the expression for \(\phi\) to obtain: \(w = \frac{d}{dt}(at + bt^{3} - ct^{4}) = a + 3bt^{2} - 4ct^{3}\).
02

Compute the derivative of the angular velocity to find the angular acceleration

The angular acceleration \(\alpha\) is the derivative of the angular velocity \(w\) with respect to time: \(\alpha = \frac{dw}{dt}\). Therefore, differentiate the expression for \(w\) to obtain: \(\alpha = \frac{d}{dt}(a + 3bt^{2} - 4ct^{3}) = 6bt - 12ct^{2}\).

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Most popular questions from this chapter

Starting from rest at \(t=0\), a wheel undergoes a constant angular acceleration. When \(t=2.33 \mathrm{~s}\), the angular velocity of the wheel is \(4.96 \mathrm{rad} / \mathrm{s}\). The acceleration continues until \(t=23.0 \mathrm{~s}\), when it abruptly ceases. Through what angle does the wheel rotate in the interval \(t=0\) to \(t=46.0 \mathrm{~s}\) ?

A pulley wheel \(8.14 \mathrm{~cm}\) in diameter has a \(5.63\) -m-long cord wrapped around its periphery. Starting from rest, the wheel is given an angular acceleration of \(1.47 \mathrm{rad} / \mathrm{s}^{2} .\) (a) Through what angle must the wheel turn for the cord to unwind? \((b)\) How long does it take?

If an airplane propeller of radius \(5.0 \mathrm{ft}(=1.5 \mathrm{~m})\) rotates at 2000 rev/min and the airplane is propelled at a ground speed of \(300 \mathrm{mi} / \mathrm{h}(=480 \mathrm{~km} / \mathrm{h})\), what is the speed of a point on the tip of the propeller, as seen by \((a)\) the pilot and \((b)\) an observer on the ground? Assume that the plane's velocity is parallel to the propeller's axis of rotation.

A planet \(P\) revolves around the Sun in a circular orbit, with the Sun at the center, which is coplanar with and concentric to the circular orbit of Earth \(E\) around the Sun. \(P\) and \(E\) revolve in the same direction. The times required for the revolution of \(P\) and \(E\) around the Sun are \(T_{P}\) and \(T_{E}\). Let \(T_{S}\) be the time required for \(P\) to make one revolution around the Sun relative to \(E\) : show that \(1 / T_{S}=1 / T_{E}-1 / T_{P}\). Assume \(T_{P}>T_{E}\).

Show that 1 rev \(/ \mathrm{min}=0.105 \mathrm{rad} / \mathrm{s}\).
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