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KE rot = 1 2 2 . size 12{"KE" rSub { size 8{"rot"} } = { {1} over {2} } Iω rSup { size 8{2} } } {}

The expression for rotational kinetic energy is exactly analogous to translational kinetic energy, with I size 12{I} {} being analogous to m size 12{m} {} and ω size 12{ω} {} to v size 12{v} {} . Rotational kinetic energy has important effects. Flywheels, for example, can be used to store large amounts of rotational kinetic energy in a vehicle, as seen in [link] .

The figure shows a bus carrying a large flywheel on its board in which rotational kinetic energy is stored.
Experimental vehicles, such as this bus, have been constructed in which rotational kinetic energy is stored in a large flywheel. When the bus goes down a hill, its transmission converts its gravitational potential energy into KE rot size 12{ ital "KE" rSub { size 8{ ital "rot"} } } {} . It can also convert translational kinetic energy, when the bus stops, into KE rot size 12{ ital "KE" rSub { size 8{ ital "rot"} } } {} . The flywheel’s energy can then be used to accelerate, to go up another hill, or to keep the bus from going against friction.

Calculating the work and energy for spinning a grindstone

Consider a person who spins a large grindstone by placing her hand on its edge and exerting a force through part of a revolution as shown in [link] . In this example, we verify that the work done by the torque she exerts equals the change in rotational energy. (a) How much work is done if she exerts a force of 200 N through a rotation of 1.00 rad ( 57.3º ) size 12{1 "." "00"`"rad" \( "57" "." 3 \) rSup { size 8{ circ } } } {} ? The force is kept perpendicular to the grindstone’s 0.320-m radius at the point of application, and the effects of friction are negligible. (b) What is the final angular velocity if the grindstone has a mass of 85.0 kg? (c) What is the final rotational kinetic energy? (It should equal the work.)

Strategy

To find the work, we can use the equation net W = net τ θ size 12{"net "W= left ("net "τ right )θ} {} . We have enough information to calculate the torque and are given the rotation angle. In the second part, we can find the final angular velocity using one of the kinematic relationships. In the last part, we can calculate the rotational kinetic energy from its expression in KE rot = 1 2 2 size 12{"KE" rSub { size 8{"rot"} } = { {1} over {2} } Iω rSup { size 8{2} } } {} .

Solution for (a)

The net work is expressed in the equation

net W = net τ θ , size 12{"net "W= left ("net "τ right )θ} {}

where net τ size 12{τ} {} is the applied force multiplied by the radius ( rF ) size 12{ \( ital "rF" \) } {} because there is no retarding friction, and the force is perpendicular to r size 12{r} {} . The angle θ size 12{θ} {} is given. Substituting the given values in the equation above yields

net W = rF θ = 0.320 m 200 N 1.00 rad = 64.0 N m.

Noting that 1 N · m = 1 J ,

net W = 64.0 J . size 12{"net "W="64" "." 0" J"} {}
The figure shows a large grindstone of radius r which is being given a spin by applying a force F in a counterclockwise direction, as indicated by the arrows.
A large grindstone is given a spin by a person grasping its outer edge.

Solution for (b)

To find ω size 12{ω} {} from the given information requires more than one step. We start with the kinematic relationship in the equation

ω 2 = ω 0 2 + 2 αθ . size 12{ω rSup { size 8{2} } =ω rSub { size 8{0} rSup { size 8{2} } } +2 ital "αθ"} {}

Note that ω 0 = 0 size 12{ω rSub { size 8{0} } =0} {} because we start from rest. Taking the square root of the resulting equation gives

ω = 2 αθ 1 / 2 . size 12{ω= left (2 ital "αθ" right ) rSup { size 8{1/2} } } {}

Now we need to find α size 12{α} {} . One possibility is

α = net τ I , size 12{α= { {"net "τ} over {I} } } {}

where the torque is

net τ = rF = 0.320 m 200 N = 64.0 N m . size 12{"net "τ= ital "rF"= left (0 "." "320"" m" right ) left ("200"" N" right )="64" "." 0" N" cdot m} {}

The formula for the moment of inertia for a disk is found in [link] :

I = 1 2 MR 2 = 0.5 85.0 kg 0.320 m 2 = 4.352 kg m 2 . size 12{I= { {1} over {2} } ital "MR" rSup { size 8{2} } =0 "." 5 left ("85" "." 0" kg" right ) left (0 "." "320"" m" right ) rSup { size 8{2} } =4 "." "352"" kg" cdot m rSup { size 8{2} } } {}

Substituting the values of torque and moment of inertia into the expression for α size 12{α} {} , we obtain

α = 64 . 0 N m 4.352 kg m 2 = 14.7 rad s 2 . size 12{α= { {"64" "." "0 N" cdot m} over {4 "." "352"" kg" cdot m rSup { size 8{2} } } } ="14" "." 7 { {"rad"} over {s rSup { size 8{2} } } } } {}

Now, substitute this value and the given value for θ size 12{θ} {} into the above expression for ω size 12{ω} {} :

ω = 2 αθ 1 / 2 = 2 14.7 rad s 2 1.00 rad 1 / 2 = 5.42 rad s . size 12{ω= left (2 ital "αθ" right ) rSup { size 8{1/2} } = left [2 left ("14" "." 7 { {"rad"} over {s rSup { size 8{2} } } } right ) left (1 "." "00"" rad" right ) right ] rSup { size 8{1/2} } =5 "." "42" { {"rad"} over {s} } } {}

Solution for (c)

The final rotational kinetic energy is

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Source:  OpenStax, Mechanics. OpenStax CNX. Apr 15, 2013 Download for free at http://legacy.cnx.org/content/col11506/1.2
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