Question

The angular speed of an automobile engine is increased uniformly from 1170 rev/min to 2880 rev/min in \(12.6 \mathrm{~s}\). (a) Find the angular acceleration in rev/min \(^{2} .(b)\) How many revolutions does the engine make during this time?

Short answer

a) \(\alpha' = 2.76 \ rev/min²\). b) Total revolutions \(N = 585 \ revolutions\).

Step-by-step solution

Convert Initial and Final Speeds to rev/sec

Since the time given is in seconds but the speeds are in rev/min, it's crucial to convert these speeds to rev/sec. This is done by dividing the given speeds by 60. Therefore, the Initial speed, \(\omega_i = 1170/60 \ rev/sec\)and Final speed, \(\omega_f = 2880/60 \ rev/sec\)

Calculate Angular Acceleration

Angular acceleration \(\alpha\) can be found by using the formula \(\alpha = (\omega_f - \omega_i) / t\). Plugging the values from Step 1 and given time t= 12.6 sec into the equation gives: \(\alpha = (\omega_f - \omega_i) / t\)

Convert Angular Acceleration to rev/min²

The computed angular acceleration is in rev/sec². Since the question asks for the answer in rev/min², multiply the answer from step 2 by \(60^2\) to convert sec² to min². \(\alpha' = \alpha * 60^2 \ rev/min²\)

Calculate Total Revolutions

Number of revolutions is the integral of angular speed, which can be calculated using the formula \( N = (\omega_i + \omega_f) / 2 * t\). Substituting necessary values from Step 1 and given time into this equation gives the result for total revolutions.

Key concepts

Uniform Angular Acceleration

When we talk about uniform angular acceleration, we're referring to the constant rate at which the angular velocity of an object changes with time. Specifically, it's the change in angular speed per unit time. Something is said to have uniform angular acceleration when this rate does not change over the period of acceleration. For example, if a spinning wheel goes from 10 revolutions per minute to 50 revolutions per minute in 5 seconds, and the rate of change in speed is constant throughout those 5 seconds, the wheel has uniform angular acceleration.

The formula to calculate angular acceleration is: \[ \alpha = \frac{\Delta \omega}{\Delta t} \] where \(\alpha\) is the angular acceleration, \(\Delta \omega\) is the change in angular velocity, and \(\Delta t\) is the time it takes for the change to occur. In our exercise, the angular acceleration is kept constant between the initial and final angular speeds, making it a case of uniform angular acceleration.

Understanding this concept is crucial because it allows us to predict how rotating objects will behave over time under a constant angular acceleration, similar to how one might use constant linear acceleration to predict the motion of a car speeding up on a straight path.

Angular Speed

Angular speed refers to how fast an object rotates or revolves relative to another point, usually the center of rotation. It's the angle an object rotates through in a unit of time. We commonly measure it in degrees per second, radians per second, or revolutions per minute (rpm).

In the context of our exercise, we are given an engine's angular speed in revolutions per minute. To find out how much the engine's angular speed changes every second, we converted the given units from rev/min to rev/sec by dividing by 60. This is important because in physics, most equations and calculations are based on the standard International System of Units, which usually measures time in seconds.

To capture this, we describe the angular speed as:\[ \omega = \frac{\theta}{t} \] where \(\omega\) is the angular speed, \(\theta\) is the angle rotated in radians, and \(t\) is the time taken.

Angular speed is a vector quantity, which means it has both a magnitude (how fast) and a direction (which way the object is spinning, clockwise or counterclockwise). It's a vital concept when describing rotational motion, whether you're looking at a spinning DVD, a rotating planet, or the wheels of a car.

Revolutions per Minute

Revolutions per minute (rpm) is a unit of rotational speed. It quantifies the number of full rotations completed by an object around an axis in one minute. This unit is widely used as it's usually easier to measure the number of times an object fully spins around, rather than calculating radians or degrees of rotation.

In automotive engineering, for instance, the rotational speed of an engine is typically expressed in rpm. It provides a clear indicator of how fast the engine is running, which can be critical for performance and efficiency considerations.

To relate this to our exercise, we initially receive the engine's speed in revolutions per minute, which is a familiar and practical measurement for many real-world applications. But to compute angular acceleration, we convert it to revolutions per second (rev/sec) because our time variable is in seconds. In the end, the final angular acceleration asked for in the problem is converted back to the perhaps more intuitive rev/min².

Rpm is such an important unit because it bridges the gap between the abstract world of angular mechanics and practical, observable phenomena. It helps in connecting theoretical physics with everyday observations and mechanical operations.