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Chapter 5: Problem 582

An automobile engine develops \(100 \mathrm{kw}\) when rotating at a speed of 1800 r.p.m. what torque does it deliver? \(\\{\mathrm{A}\\} 350 \mathrm{Nm} \quad\\{\mathrm{B}\\} 440 \mathrm{Nm}\) \\{C \(\\} 531 \mathrm{Nm} \quad\\{\mathrm{D}\\} 628 \mathrm{Nm}\)

Short Answer

Expert verified
The short answer is: The torque delivered by the engine is approximately \(531\,\mathrm{Nm}\) (Option \(\mathrm{C}\)).

Step by step solution

01

Convert RPM to radians per second

Before we can use the given formula to calculate torque, we need to convert the rotational speed from RPM (revolutions per minute) to radians per second. First, we should know that there are 2π radians in one revolution. Also, 1 minute has 60 seconds. Therefore, to convert 1800 RPM to radians per second: \[Angular\,Speed (rad/s) = 1800\,RPM × \frac{2π\,rad}{1\,revolution} × \frac{1\,min}{60\,s}\]
02

Calculate Angular Speed

To find the angular speed in radians per second, plug the values in the conversion equation: \[Angular\,Speed (rad/s) = 1800 × \frac{2π}{1} × \frac{1}{60}\] \[Angular\,Speed (rad/s) = 1800 × \frac{π}{30}\] \[Angular\,Speed (rad/s) = 60π\,rad/s\]
03

Use the power-torque formula

Now, we will use the formula that relates power, torque, and angular speed to find the torque: \[Power = Torque × Angular\,Speed\] Next, we need to rearrange the formula to solve for torque: \[Torque = \frac{Power}{Angular\,Speed}\]
04

Calculate the torque

Now, we can calculate the torque by plugging in the given power and the calculated angular speed from Step 2: \[Torque = \frac{100\,\mathrm{kW}}{60π\,\mathrm{(rad/s)}}\] We first need to convert power from kilowatts (kW) to watts (W) by multiplying by 1000: \[Torque = \frac{100,000\,\mathrm{W}}{60π\,\mathrm{(rad/s)}}\] Now divide the power by angular speed: \[Torque ≈ \frac{100,000}{188.496}\] \[Torque ≈ 530.518\,\mathrm{Nm}\]
05

Choose the correct option

Since the calculated torque is approximately 530.518 Nm, the closest answer to this value is \(\mathrm{C:}\) 531 Nm. So the correct option is \(\mathrm{C:}\) 531 Nm.

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Most popular questions from this chapter

A wheel rotates with a constant acceleration of $2.0 \mathrm{rad} / \mathrm{sec}^{2}$ If the wheel start from rest. The number of revolution it makes in the first ten seconds will be approximately. \(\\{\mathrm{A}\\} 8\) \\{B \\} 16 \(\\{\mathrm{C}\\} 24\) \(\\{\mathrm{D}\\} 32\)

A solid cylinder of mass \(\mathrm{M}\) and \(\mathrm{R}\) is mounted on a frictionless horizontal axle so that it can freely rotate about this axis. A string of negligible mass is wrapped round the cylinder and a body of mass \(\mathrm{m}\) is hung from the string as shown in figure the mass is released from rest then The angular speed of cylinder is proportional to \(\mathrm{h}^{\mathrm{n}}\), where \(\mathrm{h}\) is the height through which mass falls, Then the value of \(n\) is \(\\{\mathrm{A}\\}\) zero \(\\{\mathrm{B}\\} 1\) \(\\{\mathrm{C}\\}(1 / 2)\) \([\mathrm{D}] 2\)

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