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Area moments of inertia for a filled semicircle with respect to a horizontal line passing through the centroid of the area

The second moment of area, also known as moment of inertia of plane area, area moment of inertia, polar moment of area or second area moment, is a ... more

Angular Acceleration

Torque, moment, or moment of force is the tendency of a force to rotate an object about an axis, fulcrum, or pivot.
Moment of inertia is the mass ... more

Angular Momentum

In physics, angular momentum, moment of momentum, or rotational momentum is a measure of the amount of rotation an object has, taking into account its ... more

Worksheet 333

A typical small rescue helicopter, like the one in the Figure below, has four blades, each is 4.00 m long and has a mass of 50.0 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 of 1000 kg. (a) Calculate the rotational kinetic energy in the blades when they rotate at 300 rpm. (b) Calculate the translational kinetic energy of the helicopter when it flies at 20.0 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?


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 Er . The angular velocity ω for 1 r.p.m is

Angular velocity

and for 300 r.p.m

Multiplication

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

Moment of Inertia - Rod end

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

Multiplication

and so The rotational kinetic energy is

Rotational energy

Solution for (b)

Translational kinetic energy is defined as

Kinetic energy ( related to the object 's velocity )

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

Division

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:

Potential energy

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/

Polar Moment of Area - Annulus Cross Section

The second moment of area, also known as moment of inertia of plane area, area moment of inertia, polar moment of area or second area moment, is a ... more

Polar Moment of Area - Circular Cross Section

The second moment of area, also known as moment of inertia of plane area, area moment of inertia, polar moment of area or second area moment, is a ... more

Perpendicular axis theorem ( at moments of inertia)

Perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, ... more

Conic section (polar system and one focus on the pole and the other somewhere on the 0° ray )

conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. A conic ... more

Roll angular inertia (Automobile handling)

Automobile handling and vehicle handling are descriptions of the way wheeled vehicles perform transverse to their direction of motion, particularly during ... more

Precession (Torque-free)

Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the ... more

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