Rotational kinetic energy Definition and 9 Discussions
Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the object's moment of inertia is observed:
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{\displaystyle E_{\mathrm {rotational} }={\frac {1}{2}}I\omega ^{2}}
where
ω
{\displaystyle \omega \ }
is the angular velocity
I
{\displaystyle I\ }
is the moment of inertia around the axis of rotation
E
{\displaystyle E\ }
is the kinetic energyThe mechanical work required for or applied during rotation is the torque times the rotation angle. The instantaneous power of an angularly accelerating body is the torque times the angular velocity. For free-floating (unattached) objects, the axis of rotation is commonly around its center of mass.
Note the close relationship between the result for rotational energy and the energy held by linear (or translational) motion:
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{\displaystyle E_{\mathrm {translational} }={\frac {1}{2}}mv^{2}}
In the rotating system, the moment of inertia, I, takes the role of the mass, m, and the angular velocity,
ω
{\displaystyle \omega }
, takes the role of the linear velocity, v. The rotational energy of a rolling cylinder varies from one half of the translational energy (if it is massive) to the same as the translational energy (if it is hollow).
An example is the calculation of the rotational kinetic energy of the Earth. As the Earth has a period of about 23.93 hours, it has an angular velocity of 7.29×10−5 rad/s. The Earth has a moment of inertia, I = 8.04×1037 kg·m2. Therefore, it has a rotational kinetic energy of 2.138×1029 J.
A good example of actually using earth's rotational energy is the location of the European spaceport in French Guiana. This is within about 5 degrees of the equator, so space rocket launches (for primarily geo-stationary satellites) from here to the east obtain nearly all of the full rotational speed of the earth at the equator (about 1,000 mph, sort of a "sling-shot" benefit). This saves significant rocket fuel per launch compared with rocket launches easterly from Kennedy Space Center (USA), which obtain only about 900 mph added benefit due to the lower relative rotational speed of the earth at that northerly latitude of 28 degrees.
Part of the earth's rotational energy can also be tapped using tidal power. Additional friction of the two global tidal waves creates energy in a physical manner, infinitesimally slowing down Earth's angular velocity ω. Due to the conservation of angular momentum, this process transfers angular momentum to the Moon's orbital motion, increasing its distance from Earth and its orbital period (see tidal locking for a more detailed explanation of this process).
A carousel has the shape of a circular disc with radius 1.80 m and a mass of 300 kg. There are two people with masses of 30 and 45 kg out on the edge while carousel rotates with the angular speed 0.6 rad / s.
The people move towards the center of the carousel
Calculations show that the...
This image represents the ramp.
The first part is pretty easy.
The red part has friction, and the ball rolls down it. The blue part has no friction, and the ball climbs it only owing to the translational kinetic energy that it gained at the bottom of the red ramp, which is only a fraction of...
Homework Statement
A bar of length 2.5m and mass 5kg, whose rotation point is at its center, rotates at 5 rad/s. What is the rotational kinetic energy of the bar?
If a point mass of mass 1.5kg is added to each end of the bar, assuming the angluar velocity is the same, what is the new kinetic...
Homework Statement : [/B]
A stationary horizontal platform is free to rotate about its vertical axis. The radius of the platform is R=1.6m and its moment of inertia is 660 kgm^2. A 43 kg boy jumps on the rim of the platform with the velocity 2.2 m/s tangential to the rim. What will be the...
Homework Statement
Let g be the acceleration due to gravity at the Earth's surface and K be the rotational kinetic energy of the Earth. Suppose the Earth's radius decreases by 2%. Keeping all other quantities constant,
(a) g increases by 2% and K increases by 2%
(b) g increases by 4% and K...
Homework Statement
This problem is from the 2015 AP Physics C Mechanics free response, question 3 part b.
https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap15_frq_physics_c-m.pdf
Homework Equations
K = 1/2Iω2
U = mgh
The Attempt at a Solution
The potential energy of the bar when...
Homework Statement
A solid sphere rolls down a hemisphere from rest. Find the angle at which the sphere loses contact with the surface.
R = radius of hemisphere
a = radius of sphere
Homework Equations
ΣFr = Macm,r
N-mgcosθ = -mVcm2/(R+a)
N = mgcosθ - mvcm2/(R+a) eq. (1)
Conservation in...
Homework Statement
So I have a horizontal pulley positioned at the edge of table with a mass of .2kg hanging down from a height of .76meters, the other end of the string is attached to a wooden block of mass .25kg that when the .2kg weight is dropped the wooden block is pulled towards the...
Homework Statement
A tall, cylindrical chimney falls over when its base is ruptured. Treat the chimney as a thin rod of length 49.0 m. Answer the following for the instant it makes an angle of 32.0° with the vertical as it falls. (Hint: Use energy considerations, not a torque.)
(a) What is the...