Problem 25 - Applied Physics

Question

An engine develops \(1.50 \mathrm{~kW}\) of power at \(10, \overline{0} 00\) revolutions per \(5.00 \mathrm{~min}\). What torque is applied to the engine's crankshaft?

Step-by-Step Solution

Answer

The torque applied to the engine's crankshaft is approximately 7.16 Nm.

Step 1: Understanding Power and Torque Relationship

Power (P) and torque (τ) are related by the equation P = τ \cdot \omega, where \(\omega\) is the angular velocity in radians per second. To find the torque, we need to rearrange this equation to \(τ = \frac{P}{\omega}\).

Step 2: Converting Revolutions to Radians per Second

The engine makes 10,000 revolutions in 5 minutes. First, convert this time into seconds: 5 \times 60 = 300 seconds. Therefore, the angular velocity in revolutions per second is \frac{10,000}{300} \approx 33.33 revolutions per second. To convert this to radians per second, use the conversion factor 2\pi radians per revolution: \(\omega = 33.33 \times 2 \pi \approx 209.44\) radians per second.

Step 3: Calculating Torque from Power and Angular Velocity

Now, use the relationship \(\tau = \frac{P}{\omega}\) to find the torque. The power is given as 1.50 kW, which is equivalent to 1,500 W. Substitute the values into the rearranged formula: \(\tau = \frac{1,500}{209.44} \approx 7.16\) Nm. This is the torque applied to the engine's crankshaft.

Key Concepts

Angular Velocity

Angular velocity is a measure of how fast something rotates or spins. It tells you how much angle is covered in a certain amount of time. For engines and spinning objects, angular velocity is crucial to assess performance.

Angular velocity is often represented as \(\omega\), which is measured in radians per second. The connection between the revolutions per second and the radians per second is essential because mathematical calculations for circular motion are usually performed using radians.

Consider the following:

• 1 revolution equals \(2\pi\) radians because the circumference of a circle (one full spin) is \(2\pi\) times the radius.
• To find angular velocity in radians per second, multiply the number of revolutions per second by \(2\pi\).

Understanding this conversion allows you to calculate other related physics properties, like torque, with ease.

Power in Physics

Power in physics is the rate at which work is done or energy is transferred over time. It is a vital concept in understanding how machines and engines operate, as it determines how quickly work can be performed.

Power is often measured in watts (W), with 1 watt equaling 1 joule per second. In our original problem's context, power is given in kilowatts (kW):

• 1 kW = 1,000 W
• Thus, 1.5 kW = 1,500 W

Power is related to torque and angular velocity using the formula \(P = \tau \cdot \omega\). This equation illustrates how much power is being used to generate torque at a specific angular velocity.

By understanding and calculating power, one can better comprehend an engine's efficiency and its performance capacity.

Revolutions to Radians Conversion

Converting revolutions to radians is an essential step when working with rotational movements because most angular equations use radians.

Here's a straightforward way to convert between these measurements:

• Understand that 1 revolution is equal to \(2\pi\) radians.
• To find how many radians are in a given number of revolutions, multiply the revolutions by \(2\pi\).

For example, if an engine rotates at 33.33 revolutions per second, the conversion to radians per second is:
\[33.33 \times 2\pi \approx 209.44 \text{ radians per second}\]
This conversion is vital for calculating other physics measurements, such as torque, using the relationship between power and angular velocity.