Problem 19 - Applied Physics

Question

A flywheel rotates at 1050 rpm. (a) How long (in s) does it take to complete one revolution? (b) How many revolutions does it complete in \(5.00 \mathrm{~s}\) ?

Step-by-Step Solution

Answer

(a) 0.0571 s for one revolution; (b) 87.5 revolutions in 5 seconds.

Step 1: Convert Rotational Speed to Revolutions per Second

The flywheel rotates at 1050 revolutions per minute (rpm). To find how many revolutions it makes in one second, divide 1050 by 60 (since there are 60 seconds in one minute): \[\text{Revolutions per second} = \frac{1050}{60}\]

Step 2: Calculate Time for One Revolution

Once we know how many revolutions occur in one second, we can determine the time it takes for a single revolution by taking the reciprocal:\[\text{Time for one revolution} = \frac{1}{\text{Revolutions per second}}\]

Step 3: Calculate Total Revolutions in 5 Seconds

Now calculate how many revolutions occur in 5 seconds. Multiply the number of revolutions per second by 5:\[\text{Total revolutions in 5 seconds} = \text{Revolutions per second} \times 5\]

Key Concepts

Angular Velocity

Angular velocity is a key concept when discussing rotational motion. It describes how fast an object rotates or spins around an axis. Imagine a car tire spinning as the car moves forward; the tire has an angular velocity. Angular velocity is usually measured in units like radians per second (rad/s) or degrees per second (°/s).

• The higher the angular velocity, the faster the object spins.
• Angular velocity is a vector quantity, meaning it has both magnitude and direction.

Understanding angular velocity helps in analyzing rotational systems, such as engines, fan blades, or even amusement park rides.

Revolutions per Minute

Revolutions per minute, or rpm, is a common unit for measuring rotational speed. It tells us how many full rotations an object makes in one minute. For instance, if a flywheel spins at 1050 rpm, this speed indicates that it completes 1050 full circles every minute. Calculating rpm is straightforward. You simply count the number of rotations in a period of one minute.

• Higher rpm means more rotations per minute, implying a faster spinning object.
• Rpm is a practical unit because it aligns well with how many mechanical systems operate and how we measure time.

This unit is particularly useful in machinery and automotive industries where specific speed regulation is essential.

Time Calculation

When dealing with rotational speed and motion, it's often necessary to calculate how long it takes to complete a single revolution. This involves understanding the relationship between time, revolutions, and rotational speed.To find the time for one revolution, start with the object's rotational speed in revolutions per second (rps). Once you have this, the calculation is:

• Time for one revolution = \( \frac{1}{\text{Revolutions per second}} \)
• For instance, if a flywheel rotates at 17.5 rps (from converting 1050 rpm), each revolution takes \( \frac{1}{17.5} \) seconds.

Understanding this relationship is crucial for tasks that require precise timing and synchronization, such as in watchmaking or computer hard drives.

Revolutions

Revolutions count how many complete turns an object has made. When analyzing rotational dynamics, identifying the number of revolutions can be essential for understanding the motion of the object.Let's say you need to determine how many revolutions occur in a given time period. Multiply the time duration by the revolutions per second:

• Total revolutions = Revolutions per second \( \times \) Time (seconds)
• If a system spins at 17.5 rps, in 5 seconds, it would achieve \( 17.5 \times 5 \) revolutions, which equals 87.5 revolutions.

Knowing the number of revolutions can help you gauge the work done by rotating machinery or measure distances traveled by rotating objects like wheels.