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CHAPTER 9: ROTATIONAL MOTION, NUR ADIBAH BINTI ROZAINI - Coggle Diagram
CHAPTER 9: ROTATIONAL MOTION
8.3 - CONSTANT ANGULAR ACCELERATION
The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones.
8.5 - ROTATIONAL DYNAMICS; TORQUE AND ROTATIONAL INERTIA
From F = ma, we can see that
As the angular acceleration is the same for the whole object, we can write:
The quantity I = ΣmiRi^2 is called the rotational inertia of an object.
The distribution of mass matters here—these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation.
The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation—compare (f) and (g), for example.
8.4 - TORQUE
To make an object start rotating, a force is needed; the position and direction of the force matter as well.
The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm.
A longer lever arm is very helpful in rotating objects.
Here, the lever arm for FA is the distance from the knob to the hinge; the lever arm for FD is zero; and the lever arm for FC is as shown.
The torque is defined as:
Torque = R┴ F = RF┴.
8.1 - ANGULAR QUANTITIES
In purely rotational motion, all points on the object move in circles around the axis of rotation ("O"). The radius of the circle is R. All points on a straight line drawn through the axis move through the same angle in the same time. The angle θ in radians is defined:
θ = l/R ( l is the arc length)
Looking at a wheel that is rotating counterclockwise about an axis through the wheel’s center at O (axis perpendicular to the page). Each point, such as point P, moves in a circular path; l is the distance P travels as the wheel rotates through the angle θ.
Angular displacement.
∆θ = θ2 - θ1
The average angular velocity is defined as the total angular displacement divided by time:
The instantaneous angular velocity:
A wheel rotates from (a) initial position θ1 to (b) final position θ2. The angular displacement is Δθ = θ2 – θ1.
The angular acceleration is the rate at which the angular velocity changes with time:
The instantaneous acceleration:
Every point on a rotating body has an angular velocity ω and a linear velocity v.
They are related:
v = Rω
Objects farther from the axis of rotation will move faster.
If the angular velocity of a rotating object changes, it has a tangential acceleration:
a tan = dv/dt = Rdω/dt = Rα
Even if the angular velocity is constant, each point on the object has a centripetal acceleration:
aR = v^2/R = (Rω)^2/R = ω^2R
Here is the correspondence between linear and rotational quantities:
The frequency is the number of complete revolutions per second:
Frequencies are measured in hertz:
The period is the time one revolution takes:
8.10 - WHY DOES A ROLLING SPHERE SLOW DOWN?
Sphere rolling to the right
The frictional force has to act at the point of contact; this means the angular speed of the sphere would increase.
Gravity and the normal force both act through the center of mass, and cannot create a torque.
No real sphere is perfectly rigid. The bottom will deform, and the normal force will create a torque that slows the sphere.
The normal force, FN, exerts a torque that slows down the sphere. The deformation of the sphere and the surface it moves on has been exaggerated for detail.
8.8 - ROTATIONAL KINETIC ENERGY
The kinetic energy of a rotating object is given by
By substituting the rotational quantities, we find that the rotational kinetic energy can be written:
A object that both translational and rotational motion also has both translational and rotational kinetic energy:
When using conservation of energy, both rotational and translational kinetic energy must be taken into account.
All these objects have the same potential energy at the top, but the time it takes them to get down the incline depends on how much rotational inertia they have.
The torque does work as it moves the wheel through an angle θ:
8.9 - ROTATIONAL PLUS TRANSLATIONAL MOTION; ROLLING
In (a), a wheel is rolling without slipping. The point P, touching the ground, is instantaneously at rest, and the center moves with velocity v. The wheel rolling to the right. Its center C moves with velocity v. point P ia at rest at this instant.
In (b) the same wheel is seen from a reference frame where C is at rest. Now point P is moving with velocity -v.
The linear speed of the wheel is related to its angular speed:
v = Rω
8.6 - SOLVING PROBLEMS IN ROTATIONAL DYNAMICS
SOLVING PROBLEMS
Draw a diagram.
Decide what the system comprises.
Draw a free-body diagram for each object under consideration, including all the forces acting on it and where they act.
Find the axis of rotation; calculate the torques around it.
Apply Newton’s second law for rotation. If the rotational inertia is not provided, you need to find it before proceeding with this step.
Apply Newton’s second law for translation and other laws and principles as needed.
Solve
Check your answer for units and correct order of magnitude.
8.7 - DETERMINING MOMENTS OF INERTIA
If a physical object is available, the moment of inertia can be measured experimentally.
Otherwise, if the object can be considered to be a continuous distribution of mass, the moment of inertia may be calculated:
The parallel-axis theorem gives the moment of inertia about any axis parallel to an axis that goes through the center of mass of an object:
The perpendicular-axis theorem is valid only for flat objects.
Iz = Ix + Iy
8.2 - VECTOR NATURE OF ANGULAR ACCELERATION
The angular velocity vector points along the axis of rotation, with the direction given by the right- hand rule. If the direction of the rotation axis does not change, the angular acceleration vector points along it as well.
NUR ADIBAH BINTI ROZAINI
