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Chapter 10: Problem 45

You rev your car's engine and watch the tachometer climb steadily from 1200 rpm to 5500 rpm in 2.7 s. What are (a) the engine's angular acceleration and (b) the tangential acceleration of a point on the edge of the engine's 3.5 -cm-diameter crankshaft? (c) How many revolutions does the engine make during this time?

Short Answer

Expert verified
Angular acceleration is calculated as \(\alpha = 893.55 \, rad/s^2\). The tangential acceleration of a point on the edge of the engine's crankshaft is \(a_t = 1.5604 \, m/s^2\). The engine makes approximately 92.5 revolutions.

Step by step solution

01

Find Angular Acceleration

We first calculate the change in the angular speed, also known as Angular acceleration. As the problem informs that the accelerator’s RPM (Revolutions Per Minute) increased from 1200 to 5500, we need to find the change in rotation speed first. Convert RPM to rad/sec using the conversion factor 2π/60. The initial angular velocity, \( \omega_i = 1200 \times \frac{2\pi}{60} \) rad/s and the final angular velocity, \( \omega_f = 5500 \times \frac{2\pi}{60} \) rad/s. Angular acceleration \( \alpha \) is defined as \( \alpha = \frac{\omega_f - \omega_i}{\Delta t} \), where \( \Delta t \) is the time interval. Substitute the values into the formula to find \( \alpha \).
02

Calculate Tangential Acceleration

Next, we move on to calculate the tangential acceleration. Tangential acceleration (\(a_t\)) is given by \(a_t = \alpha \times r\), with r being the radius of the crankshaft. The diameter is given as 3.5 cm, so the radius \(r = \frac{3.5}{2} \) cm. We need to convert this to meters to match our units from step 1, so \( r = \frac{3.5 \times 10^{-2}}{2} \) m. Substitute \(r\) and \(\alpha\) from step 1 into the equation to find \(a_t\).
03

Determine the Number of Revolutions

Finally, to find the number of engine revolutions during the acceleration, we can apply the formula for angular displacement which is \(\Delta \theta = \omega_i \times \Delta t + 0.5 \times \alpha \times (\Delta t)^2\). As one revolution is equivalent to \(2\pi\) rad, divide \( \Delta \theta \) by \(2\pi\) to obtain the number of revolutions.

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