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Moment of Inertia - Right Circular Cone - x and y axis

In physics and applied mathematics, the mass moment of inertia, usually denoted by I, measures the extent to which an object resists rotational ... more

Moment of Inertia - Right Circular Cone - z axis

In physics and applied mathematics, the mass moment of inertia, usually denoted by I, measures the extent to which an object resists rotational ... more

Volume of any right circular cone

Right circular cone is assumed to be a cone that its base is a circle and right means that the axis passes through the centre of the base at right angles ... more

Volume of a cone - circular

A cone is an n-dimensional geometric shape that tapers smoothly from a base (usually flat and circular) to a point called the apex or vertex. It is the ... more

Volume of cone (by the diameter)

Description

A cone is an n-dimensional geometric shape that tapers smoothly from a base (usually flat and circular) to a point called the apex or ... more

Period of Precession - (Torque-induced - Classical Newtonian)

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

Precession - (Torque-induced - Classical Newtonian)

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

Moment of inertia of a thick-walled cylindrical tube ( Axis at the center of the cylinder perpendicular to its height)

Moment of inertia is the mass property of a rigid body that defines the torque needed for a desired angular acceleration about an axis of rotation. Moment ... more

Area Moment of Inertia - Filled Right Triangle

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

Moment of inertia of thick-walled cylindrical tube with open ends

Mass moment of inertia, measures the extent to which an object resists rotational acceleration about an axis, and is the rotational analogue to mass.
... more

Total Area of a Frustum of a Right Circular Cone

In geometry, a frustum is the portion of a solid (normally a cone or pyramid) that lies between two parallel planes cutting it.The total area of a frustum ... more

Moment of inertia of a thin rectangular plate (Axis of rotation in the center of the plate)

Mass moment of inertia measures the extent to which an object resists rotational acceleration about an axis, and is the rotational analogue to mass. Moment ... more

Moment of inertia of a thin rectangular plate (Axis of rotation at the end of the plate)

Moment of inertia is the mass property of a rigid body that determines the torque needed for a desired angular acceleration about an axis of rotation. ... more

Moment of inertia of a solid cuboid ( Axis of rotation at the height )

Moment of inertia is the mass property of a rigid body that defines the torque needed for a desired angular acceleration about an axis of rotation. Moment ... more

Moment of inertia of a solid cuboid ( Axis of rotation at the width)

Moment of inertia is the mass property of a rigid body that defines the torque needed for a desired angular acceleration about an axis of rotation. Moment ... more

Moment of inertia of a solid cuboid ( Axis of rotation at the depth)

Moment of inertia is the mass property of a rigid body that defines the torque needed for a desired angular acceleration about an axis of rotation. Moment ... more

Moment of inertia of a solid cuboid ( Axis of rotation at the longest diagonal )

oment of inertia is the mass property of a rigid body that defines the torque needed for a desired angular acceleration about an axis of rotation. Moment ... more

Area Moment of Inertia - Filled Circular Sector

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

Moment of Inertia - Sphere (shell)

In physics and applied mathematics, the mass moment of inertia, usually denoted by I, measures the extent to which an object resists rotational ... more

Moment of Inertia - Sphere (hollow)

In physics and applied mathematics, the mass moment of inertia, usually denoted by I, measures the extent to which an object resists rotational ... more

Moment of Inertia - Sphere (solid)

In physics and applied mathematics, the mass moment of inertia, usually denoted by I, measures the extent to which an object resists rotational ... more

Moment of inertia factor

In planetary sciences, the moment of inertia factor or normalized polar moment of inertia is a dimensionless quantity that characterizes the radial ... more

Darwin / Radau equation

In astrophysics, the Darwin / Radau equation gives an approximate relation between the moment of inertia factor of a planetary body and its rotational ... more

Moment of inertia of a torus of tube

Mass moment of inertia measures the extent to which an object resists rotational acceleration about an axis, and is the rotational analogue to mass.
A ... more

Moment of inertia of a torus of tube (about a diameter)

Mass moment of inertia measures the extent to which an object resists rotational acceleration about an axis, and is the rotational analogue to mass.
A ... more

Radius of oscillation of a Simple Pendulum

A pendulum is a weight suspended from a pivot (massive bob), so that it can swing freely.The length of the ideal simple pendulum is the distance from the ... 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/

Area Moment of Inertia - 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

Shaft bending moment due to yaw (2-bladed rotor)

The shaft bending moment due to yaw depends on the blades of the rotor. In this case the rotor has 2 blades

... more

Blade root bending moment load due to yaw

The blade root bending moment due to the wind turbine yaw operation. The yaw rate can be calculated for passive yaw, or is defined by the design for active ... more

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