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he formula is a special case of general Faulhaber’s formula and gives the sum of natural consecutive numbers raised to the fifth power,(starting with 1), ... more

A triangular number or triangle number counts the objects that can form an equilateral triangle. The nth triangle number is the number of dots composing a ... more

In number theory, the sum of the first n cubes is the square of the nth triangular number. The sequence of squared triangular numbers is

0, 1, 9, ... more

A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous ... more

In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square ... more

A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous ... more

An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant and is ... more

The Heronian mean of two non-negative real numbers is a weighted mean of their arithmetic and geometric means.The weighted mean is similar to an ... more

In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the ... more

Acoustic resonance is the tendency of an acoustic system to absorb more energy when it is forced or driven at a frequency that matches one of its own ... more

The weighted mean is similar to an arithmetic mean (the most common type of average), where instead of each of the data points contributing equally to the ... more

Although an exact analytical solution of the Buckingham-Reiner equation can be obtained because it is a fourth order polynomial equation in f, due to ... more

The geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of n numbers by using the product of their ... more

An I-beam, also known as H-beam, W-beam (for “wide flange”), Universal Beam (UB), Rolled Steel Joist (RSJ), or ... more

In probability theory and statistics, kurtosis is any measure of the “tailedness” of the probability distribution of a real-valued random ... more

The Stefan–Boltzmann law, also known as Stefan’s law, describes the power radiated from a black body in terms of its temperature. Specifically, the ... more

In electronics, gain is a measure of the ability of a two-port circuit (often an amplifier) to increase the power or amplitude of a signal from the input ... more

The Sherwood number (Sh) is a dimensionless number used in mass-transfer operation. It can be further defined as a function of the Reynolds and Schmidt ... more

In respiratory physiology, airway resistance is the resistance of the respiratory tract to airflow during inspiration and expiration. In fluid dynamics, ... more

An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant and is ... more

The amount of electrical charge that must be added to an isolated conductor to raise its electrical potential by one unit

... more

Rayleigh scattering (pronounced /ˈreɪli/ RAY-lee), named after the British physicist Lord Rayleigh (John William Strutt), is the (dominantly) elastic ... more

Dermott’s law is an empirical formula for the orbital period of major satellites orbiting planets in the Solar System. It was identified by the ... more

A Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous fluid at high stress. It is named after ... more

A Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous fluid at high stress. It is named after ... more

In mathematics, generalised means are a family of functions for aggregating sets of numbers, that include as special cases the arithmetic, geometric, and ... more

The first image shows how helicopters store large amounts of rotational kinetic energy in their blades. This energy must be put into the blades before takeoff and maintained until the end of the flight. The engines do not have enough power to simultaneously provide lift and put significant rotational energy into the blades.

The second image shows a helicopter from the Auckland Westpac Rescue Helicopter Service. Over 50,000 lives have been saved since its operations beginning in 1973. Here, a water rescue operation is shown. (credit: 111 Emergency, Flickr)

Strategy

Rotational and translational kinetic energies can be calculated from their definitions. The last part of the problem relates to the idea that energy can change form, in this case from rotational kinetic energy to gravitational potential energy.

Solution for **(a)**

We must convert the angular velocity to radians per second and calculate the moment of inertia before we can find **E _{r}** . The angular velocity

**ω**for

**1 r.p.m**is

and for **300 r.p.m**

The moment of inertia of one blade will be that of a thin rod rotated about its end.

The total I is four times this moment of inertia, because there are four blades. Thus,

and so The rotational kinetic energy is

Solution for **(b)**

Translational kinetic energy is defined as

To compare kinetic energies, we take the ratio of translational kinetic energy to rotational kinetic energy. This ratio is

Solution for **(c)**

At the maximum height, all rotational kinetic energy will have been converted to gravitational energy. To find this height, we equate those two energies:

Discussion

The ratio of translational energy to rotational kinetic energy is only **0.380**. This ratio tells us that most of the kinetic energy of the helicopter is in its spinning blades—something you probably would not suspect. The **53.7 m** height to which the helicopter could be raised with the rotational kinetic energy is also impressive, again emphasizing the amount of rotational kinetic energy in the blades.

Reference : OpenStax College,College Physics. OpenStax College. 21 June 2012.

http://openstaxcollege.org/textbooks/college-physics

Creative Commons License : http://creativecommons.org/licenses/by/3.0/

Although an exact analytical solution of the Buckingham-Reiner equation can be obtained because it is a fourth order polynomial equation in f, due to ... more

The Richardson number (Ri) is named after Lewis Fry Richardson (1881 – 1953). It is the dimensionless number that expresses the ratio of potential to ... more

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A typical small rescue helicopter, like the one in the Figure below, has four blades, each is

4.00 mlong and has a mass of50.0 kg. The blades can be approximated as thin rods that rotate about one end of an axis perpendicular to their length. The helicopter has a total loaded mass of1000 kg.(a)Calculate the rotational kinetic energy in the blades when they rotate at300 rpm.(b)Calculate the translational kinetic energy of the helicopter when it flies at20.0 m/s, and compare it with the rotational energy in the blades.(c)To what height could the helicopter be raised if all of the rotational kinetic energy could be used to lift it?