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Chapter 3: Problem 83

A flywheel rotates at constant speed of \(3000 \mathrm{rpm}\). The angle described by the shaft in radian in one second is (a) \(2 \pi\) (b) \(30 \pi\) (c) \(100 \pi\) (d) \(3000 \pi\)

Short Answer

Expert verified
The angle described by the shaft in radian in one second is \(100 \pi\) rad/sec. Therefore, the correct option is (c) \(100 \pi\).

Step by step solution

01

Convert rpm to revolutions per second

Firstly, we must convert the given rotational speed from revolutions per minute (rpm) to revolutions per second. Given \(3000 \, rpm\), we can divide by \(60 \, sec/min\) to convert it to revolutions per second. Therefore we get: \(3000 \, rpm \times \frac{1 \, min}{60 \, sec} = 50 \, rev/sec\).
02

Convert revolutions to radians

Each revolution is equivalent to \(2 \pi\) radians, therefore we multiply the speed in revolutions per second by \(2 \pi \, rad/rev\). The result is \(50 \, rev/sec \times 2 \pi \, rad/rev = 100 \pi \, rad/sec\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Angular Velocity
Angular velocity is a key concept in rotational mechanics, which describes how quickly an object rotates around a specific axis. It is analogous to linear velocity, but instead of moving through space in a line, it involves circular motion. Angular velocity is often denoted by the Greek letter omega (\(\omega\)).

Let's break it down:
  • Angular velocity measures the rate of change of the angular position of an object. It tells us how fast something is spinning.
  • The unit for angular velocity is radians per second, but it can also be expressed in revolutions per minute (rpm) depending on the context.
  • To find the angular velocity, you need to determine the change in angle over time. This is often given in terms of revolutions, which are then converted to radians.
By understanding angular velocity, we can predict and calculate the motion of rotating objects, from a spinning flywheel to the rotation of planets in space. It's a fundamental concept in physics and engineering, providing insights into the behavior of rotating systems.
What are Radians per Second?
Radians per second is a unit used to express angular velocity. It's essential to grasp the concept of radians first:
  • A radian is a measure of angle based on the radius of a circle. It's the angle created when the length along the arc of the circle is equal to the circle's radius.
  • There are \(2\pi\) radians in a full circle, equivalent to 360 degrees.

When we talk about radians per second, we're discussing how many radians an object sweeps out in one second as it rotates.
  • For example, if an object has an angular velocity of \(100\pi\) radians per second, it rotates through \(100\pi\) radians in one second.
  • This unit is very useful because it directly relates the rotational speed of an object to its physical motion.
Understanding radians per second helps you to directly link the mathematical description of rotation with the physical movement of an object.
Revolutions per Minute (RPM) Explained
Revolutions per minute (rpm) is another common unit of angular velocity. It's often used in everyday contexts like car engines or flywheels.
  • RPM measures how many complete turns an object makes around its axis in one minute.
  • It's a straightforward way to quantify the speed of rotation, especially when dealing with mechanical systems.
  • For example, in automotive contexts, knowing the engine's rpm can help determine the best gear to use for efficiency.

Converting from rpm to radians per second involves a straightforward calculation. Since one revolution is \(2\pi\) radians and there are 60 seconds in a minute, you convert by multiplying the rpm by \(\frac{2\pi}{60}\) to get the angular velocity in radians per second.
  • The original exercise showed this conversion where \(3000\, rpm\) became \(100\pi\, rad/sec\).
By understanding rpm and its conversions, you can easily switch between different units of angular velocity based on the situation or problem you are dealing with.

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Most popular questions from this chapter

The angular velocity of a particle is given by \(\omega=1.5 t-3 t^{2}+2\), the time when its angular acceleration ceases to be zero will be (a) \(25 \mathrm{sec}\) (b) \(0.25 \mathrm{sec}\) (c) \(12 \mathrm{sec}\) (d) \(1.2 \mathrm{sec}\)

If mass speed and radius of rotation of a body moving in a circular path are all increased by \(50 \%\), the necessary force required to maintain the body moving in the circular path will have to be increased by (a) \(225 \%\) (b) \(125 \%\) (c) \(150 \%\) (d) \(100 \%\)

A ball rolls off top of a staircase with a horizontal velocity \(u \mathrm{~m} / \mathrm{s}\). If the steps are \(h\) metre high and \(b\) mere wide, the ball will just hit the edge of nth step if \(n\) equals to (a) \(\frac{h u^{2}}{g b^{2}}\) (b) \(\frac{u^{2} 8}{g b^{2}}\) (c) \(\frac{2 h u^{2}}{g b^{2}}\) (d) \(\frac{2 u^{2} g}{h b^{2}}\)

A bucket tied at the end of a \(1.6 \mathrm{~m}\) long string is whirled in a vercar cmere min constant sped. What should be the minimum speed so that the water from the bucket does not spill, when the bucket is at the highest position (Take \(g=10 \mathrm{~m} / \mathrm{sec}^{2}\) ) (a) \(4 \mathrm{~m} / \mathrm{sec}\) (b) \(6.25 \mathrm{~m} / \mathrm{sec}\) (c) \(16 \mathrm{~m} / \mathrm{sec}\) (d) None of these

A bomb is dropped on an enemy post by an aeroplane flying with a horizontal velocity of \(60 \mathrm{~km} / \mathrm{hr}\) and at a height of \(490 \mathrm{~m}\). How far the aeroplane must be from the enemy post at time of dropping the bomb, so that it may directly hit the target. \(\left(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\right)\) (a) \(\frac{100}{3} \mathrm{~m}\) (b) \(\frac{500}{3} \mathrm{~m}\) (c) \(\frac{200}{3} \mathrm{~m}\) (d) \(\frac{400}{3} \mathrm{~m}\)
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