What is Rotational kinetic energy: Definition and 140 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).
If there is a net external torque on the system about ##S##, then this net torque vector must have its direction on the ##z##-axis (since the system is only rotating about this axis at all times).
This torque generates angular acceleration
$$\tau_{S,z}=I_S\alpha_z\tag{1}$$
where ##I_S## is...
I am wondering if it is possible to calculate either the Kinetic Energy or Rotational Kinetic Energy of an object if we have the Power (kW), Torque (Nm), and Speed (RPM) of the object.
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...
Hello!
I was reading two things:
1) tidal locking (as explained in the Wikipedia article:https://en.wikipedia.org/wiki/Tidal_locking
where it is stated that, because of internal friction caused by the body of water being attracted to the moon and deforming, the kinetic energy of the system...
I first tried to get the solution by conserving the rotational kinetic energy and got ##\omega'=\frac2{\sqrt5} \omega##.
But, it was not the correct answer. Next I tried by conserving the angular momentum and got ##\omega'=\frac 45 \omega##, which is the correct answer.
Why is the rotational...
I was going over the rolling disk versus rolling hoop problem, in which the hoop has more Kr due to greater I and therefore smaller Kt and v. I know this can be algebraically proved with two unique expressions for V that don't involve omega. The question in class that came up concerns torque. If...
For parts A and B I used energy to find the vcom and omega, but that won’t work for C. I have an answer by combining the three formulas that use acceleration above. My answer for alpha=-5g/3r. The next two are easily solvable if you find C, but I still feel like I’m missing something. Any help...
Homework Statement
A drum major twirls a 94-cm-long, 500 g baton about its center of mass at 150 rpm. What is the baton's rotational kinetic energy?
Homework Equations
KE_rot = 0.5 I w^2
The Attempt at a Solution
w = (150rpm*2pi)/60 = 15.708rad/s
I = (1/12)*0.5kg * 0.94m = 0.039167kg*m^2...
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
Homework Equations
L = IW
RKE = .5IW^2
The Attempt at a Solution
Since the two solid spheres have equal radii and same angular momenta, except that the mass differs, i think that the inertia is equal. However, since sphere A is more heavy, it would spin slower than sphere...
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...
A mass m is attached at the end of the string. The mass moves on a frictionless table, and the string passes through a hole in the table, under which someone is pulling on the string to make it taut at all times. Initially , the mass moves in a circle, with kinetic energy [E][/0] . The string is...
If rotational kinetic energy of a closed system decreases, another form of energy must increase for the conservation of energy of a closed system.
We assume this system (a person in rotation) has these forms of energy:
ΔE=ΔK (only rotational around an axis) + ΔUi (internal) + ΔEt (thermal) with...
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 [/B]
An ice skater executes a spin about a vertical axis with her feet on a frictionless ice surface. In each hand she holds a small 5kg mass of which are both 1m from the rotation axis and the angular velocity of the skater is 10rad/s. The skater then moves her arms so that...
Homework Statement
A disk of mass m1 is rotating freely with constant angular speed ω. Another disk of mass m2 that has the same radius is gently placed on the first disk. If the surfaces in contact are rough so that there is no slipping between the disks, what is the fractional decrease in the...
I am having trouble wrapping my head around a physics concept.
If we roll solid sphere down an inclined plane it will reach the bottom at a different time than if we were to say, roll a hoop down the same inclined plane. This is because they have different rotational inertias, and so more of...
Homework Statement
I am trying to get the hamiltonain operator equality for a rigid rotor. But I don't get it. Please see the red text in the bottom for my direct problem. The rest is just the derivation I used from classical mechanics.
Homework Equations
By using algebra we obtain:
By...
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
If I have a ball moving in a circular path (ball is connected to a string), as shown in this picture:
http://w3.shorecrest.org/~Lisa_Peck/Physics/syllabus/mechanics/circularmotion/Images/cent_force_on_ball.gif
should I say that the energy of the ball is both its kinetic...
I tried to use this formula: KE=(1/2)(I)(w)^2, and work=change in KEFor the first question, i tried to plug the number into the rotation kinetic energy formula:
3600rpm=376.8rad/s
(0.5)(2,000kg)(0.125m)^2(376.8rad/s)^2
I found out this was the answer 1.1 x 106 J, but I got 2218410J
For 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...
Would there be any way to get the energy from the rotation of a planet, which is an Earth like planet in material, sized like Jupiter, and seems to have a 24 hr day.
I am just starting to get into real Physics, So I am not even sure where to start getting the numbers to work with, and I would...
Homework Statement
A motor imposes 5 Nm on a shaft of a 100kg flywheel which is 80cm(diameter)
What is the rotational kinetic energy after 10 mins? assuming the flywheel was at rest and the mass moment of inertia of the shaft is negligble.
Homework Equations
Ig=1/2 * mr^2...
Homework Statement
Consider a solid disk of mass m and radius R rolling along a horizontal surface with the center of the disk moving horizontally with a speed v. The total kinetic energy of the disk is a combination of translational kinetic energy , KEt and rotational kinetic energy, KEr...
Homework Statement
The rotating drum on a clothes washer has a rotational inertia of 1.2 kg*m^2. In the spin cycle, it rotates at 240 rev/min. (a) What length of time is required for a 0.25 hp motor to bring the drum to this rotation rate starting form rest? (b) If the angular acceleration is...
Homework Statement
A 240g ball and a 570g ball are connected by a 48.0-cm-long massless, rigid rod. The structure rotates about its center of mass at 110 rpm.
Homework Equations
KE = Iω^2
I = 1/12(mr^2)
The Attempt at a Solution
Since it has two masses and two different radius...
Homework Statement
Hello,
I am a bit confused on when rotational kinetic energy exists and when linear kinetic energy exists. For example when we spin a string with a ball attached to the end it has kinetic energy of 0.5mv^2 of the ball when we were solving for velocity of the ball at certain...
Homework Statement
A solid cylinder is rolling along a horizontal plane and is friction less around its symmetric axis. The cylinder is pulled by a constant force, F and travels the distance, d. The cylinder does not glide and has a friction force, f on the ground.
Known values:
Mass: M...
Homework Statement
A 1.80m long pole is balanced vertically with its tip on the ground. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [hint: Use conservation of energy]
l=1.890m
Homework...
Homework Statement
This is not a homework problem. Okay, it is -- but I have solved it correctly already, so the question is not there. I'm just not sure about a detail in one of many solutions.
We have a hollow cylinder with a uniform mass distribution rolling down an incline with some...
Homework Statement
A spherical object with moment of inertia 0.57mr2 rolls without slipping down an incline. At the bottom of the incline.
Homework Equations
For a sphere (or cylinder) rolling without slipping ω = v/r.
Rotational KE = ½Iω²
The translational kinetic energy = ½mv²...
Homework Statement
This question has two parts; the first part I understand but the second part I do not.
Part 1: What is the rotational energy of a planet about its spin axis? Model the planet as a uniform sphere of radius 6420 km, and mass 5.59x10^24 kg. Assume it has a rotational period...
What happens if a ball goes from rolling on a surface to a frictionless surface? I know friction is required for the ball to roll, so would the rotational energy be transformed into translational energy (the ball goes faster without rolling), or does something else happen?
Thanks!
Homework Statement
A solid cylinder is rolling without slipping. What fraction of its kinetic energy is linear?
Homework Equations
Ke=\frac{mv^2}{2}+\frac{I(v/r)^2}{2}
The Attempt at a SolutionKe=\frac{mv^2}{2}+\frac{(v/r)^2}{2}*\frac{(mr)^2}{2}
Ke=\frac{3(mv)^2}{4}
Linear Ke...
Homework Statement
A disk with a mass of 4kg and a radius of 2m is free to rotate around an axis that passes through the center of the disk and perpendicular to the plane of the disk. The rotational kinetic energy of the disk is increasing at a constant rate of 20 J/rad; that is, the energy...
I have two rigid bodies floating in space that are kinematically constrained by a joint (think of a 2 dof link mechanism floating in space).
I have a body fixed reference frame on each rigid body plus the global space-fixed reference frame. The first rigid body is in the space-fixed...
Homework Statement
If Inga, the Laboratory Assistant, rolls a spare head down a 4 m ramp because it was spherical and solid and too heavy at 4.5 kg at a speed of 4.5 m/s, what was its total kinetic energy?
Homework Equations
KE = (1/2) I ω^2
I = (2/5) MR^2
The Attempt at a Solution...
The problem I am working on is:
"Big Ben, the Parliament tower clock in London, has an hour hand 2.70 m long with a mass of 300 kg, and a minute hand 4.20 m long with a mass of 100 kg (see figure below). Calculate the total rotational kinetic energy of the two hands about the axis of rotation...
Homework Statement
A 0.125kg basketball is rolling w/out slipping on a horizontal table at 4.50 m/s when it rolls off the edge and it falls to the floor, 1.10 m below. What is the rotational kinetic energy of the ball right before hitting floor?Homework Equations
KE rot: .5 I w^2
KE...
The disc brakes of a high performance car are often made of carbon fiber instead of iron, thereby reducing the mass. If both types of discs are of the same size and shape, and each iron disc has a mass of 4 kg and each carbon disc has a mass of 1 kg, what is the reduction in rotational kinetic...
Homework Statement
A thin, cylindrical rod = 27.0 cm long with a mass m = 1.20 kg has a ball of diameter d = 10.00 cm and mass M = 2.00 kg attached to one end. The arrangement is originally vertical and stationary, with the ball at the top as shown in the figure below. The combination is...
Homework Statement
In a crude model of a rotating diatomic molecule of chlorine (Cl2), the two Cl atoms are 2.00 10^-10 m apart and rotate about their center of mass with angular speed ω = 3.80 10^12 rad/s. What is the rotational kinetic energy of one molecule of Cl2, which has a molar mass...
suppose we are sitting on earth.now we threw a object with velosity greater than escape velosity.will the kinetic energy remain same if we throw the object in different directions?how
This is problem from a textbook whose solution I don't understand.
A small rock of mass 0.5kg is attached to a 1.5m string, then it is whirled for 5 seconds until it achieves a near horizontal orbit at 120 rpm. What is the torque required?
I used the equation : τ=Iα (moment of inertia x...