If we push on an object in the forward direction while the object is moving forward, we do positive work on the object. The object accelerates, because we are pushing on it. F = ma. The object gains kinetic energy. The translational kinetic energy of an object with mass m, whose center of mass is moving with speed v is K = ½mv2.
Translational kinetic energy = ½ mass * speed2
Kinetic energy increases quadratically with speed. When the speed of a car doubles, its energy increases by a factor of four.
A rotating object also has kinetic energy. When an object is rotating
about its center of mass, its rotational kinetic
energy is K = ½Iω2.
Rotational kinetic energy = ½ moment of inertia * (angular speed)2.
When the angular velocity of a spinning wheel doubles, its kinetic energy
increases by a factor of four.
When an object has translational as well as rotational motion, we can look at
the motion of the center of mass and the motion about the center of mass
separately. The total kinetic energy is the sum of the translational kinetic energy of the center
of mass (CM) and the rotational kinetic energy about
the CM.
Consider a wheel of radius r and mass m rolling on a flat surface in the
x-direction.
The displacement Δx and the angular displacement Δθ are related through
Δx = rΔθ.
The magnitudes of the linear velocity and the angular velocity are related through vCM = rω.
The kinetic energy of the disk is the sum of the kinetic energy of the motion of the
center of mass ½mvCM2 = ½mr2ω2,
and the kinetic energy of the motion about the center of mass, ½Iω2.
Thee total kinetic energy is
KEtot = ½mr2ω2 + ½Iω2 = ½[mr2 + I]ω2 = ½[m + I/r2]v2.
Assume the wheel is a uniform disk.
The moment of inertia I of a uniform disk about an axis perpendicular to the plane of the
disk through its CM is ½mr2.
The kinetic energy of the disk therefore is KEtot = (3/4)mr2ω2.
The ratio of the translational to the rotational kinetic energy is Etrans/Erot
= mr2/I.
If two rolling object have the same total kinetic energy, then the
object with the smaller moment of inertia has the larger
translational kinetic energy and the larger speed.
Assume a disk and a ring with the same radius roll down an incline of height h and angle theta. If they both start from rest at t = 0, which one will reach the bottom first?
Solution:
Suppose you are designing a race bicycle and it comes time to work on the wheels. You are told that the wheels need to be of a certain mass but you may design them either as wheels with spokes (like traditional bike wheels) or you may make them as having solid rims all the way through. Which design would you pick given that the racing aspect of the machine is the most important? Please explain!
Discuss this with your fellow students in the discussion forum!
By clicking the button below, you can play or to step through a video clip frame-by-frame. Each step corresponds to a time interval of (1/30) s. In the clip the same torque acts on objects with different moments of inertia. The torque is the product of a weight and a small lever arm. The moment of inertia of the ruler-like object changes because masses are added at larger distances away from the center. When the weight has dropped through the same distance Δy, the same work has been done and the system has the same kinetic energy, since it starts from rest. Neglecting friction W = τΔθ = mgΔy = ½Iω2. However, the system with the larger moment of inertia I has the smaller angular speed ω. (Compare the angles through which the ruler turns per step without attached masses and with masses attached at different locations.)
Problem:Three particles are connected by rigid rods of negligible mass lying along
the y-axis as shown.
If the system rotates about the x-axis with angular speed of 2 rad/s,
find
(a) the moment of inertia about the x-axis and the total rotational
kinetic energy evaluated from ½Iω2,
and
(b) the linear speed of each particle and the total kinetic energy evaluated
from Σ½mivi2.
Solution:
Angular momentum about an axis is a measure of an objects rotational motion about this axis. For rotations about a symmetry axis of an object, the angular momentum L is defined as the product of an object's moment of inertia I times its angular velocity ω about the chosen axis.
L = Iω.
Problem:A light rod 1 m in length rotates in the xy plane about a pivot through the rod's center. Two particles of mass 4 kg and 3 kg are connected to its ends. Determine the angular momentum of the system at the instant the speed of each particle is 5 m/s.
Solution:
Angular momentum is a vector. For a single particle its direction is the direction of the angular velocity (given by the right hand rule). The angular momentum of an object is changed by giving it an angular impulse. An angular impulse ΔL is a change in angular momentum. You give an object an angular impulse by letting a torque act on it for a time interval Δt.
ΔL = τΔt
angular impulse = torque * time
If an object has many independently rotating parts, the total angular momentum of the object is the sum of the angular momenta of all its parts.
You usually give your closet door a gentle push and it swings closed gently
in 5 seconds. But today you are in a rush and exert 3 times the normal
torque on it.
(a) If you push on it for the usual time with this increased torque, how will
its angular momentum differ from the usual value?
(b) How long will it take the closet door to swing closed after your hurried
push?
Solution:
The total angular momentum of a a single object is constant if no external torque acts on the object. An object cannot exert a torque on itself. The total angular momentum of two interacting objects is also constant if no external torque acts on the objects. Newton's third law tells us the forces the objects exert on each other are equal in magnitude and opposite in direction. The interaction forces produce torques equal in magnitude and opposite in direction. These torques change the angular momentum of each object by the same amount, but the changes will have opposite directions. When we sum them up to find the change in the total angular momentum, we obtain zero.
If no external torque acts on a system of interacting objects, then their total angular momentum is constant.
In the video clip shown below the total angular momentum of the system points upward. The person is stopping a spinning wheel and the stool starts to spin.
Some of the first few frames
As the person applies a torque to the wheel, the wheel applies a torque to the person. The magnitudes of the angular momenta of the wheel and of the person change at the same rate, but their sum remains constant.
If an object has nothing external to interact with, it cannot change its angular momentum. However, a flexible object can change its angular position.
or
Please watch.
A 60 kg woman stands at the rim of a horizontal turntable having a moment
of inertia of 500 kg m2, and a radius of 2 m. The turntable is
initially at rest and is free to rotate about a frictionless vertical axis
through its center. The woman then starts walking around the rim
clockwise (as viewed from above the system) at a constant speed of 1.5 m/s
relative to the Earth.
(a) In what direction and with what angular speed does the turntable rotate?
(b) How much work does the women do to set herself and the turntable in motion?
Solution:
The total angular momentum about any axis in the universe is conserved. The angular momentum of a single object, however, changes when a net torque acts on the object for a finite time interval. Conversely, if no net torque acts on an object, then its angular momentum is constant.