Advanced Higher Physics Unit 1. Rotational Dynamics. Using moments. The spanner exert a moment or turning effect on the nut. Turning point. distance from force to turning point. force. If the moment is big enough, it will unscrew the nut.
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Advanced Higher Physics Unit 1
Rotational Dynamics
The spanner exert a moment or turning effect on the nut.
Turning point
distance from force to turning point
force
If the moment is big enough, it will unscrew the nut.
If not, there are two ways of increasing the moment.
Turning point
distance from force to turning point
force
If the same force is applied over a greater distance, a larger moment
is produced.
2. Increasing the Force applied-push/pull harder or get someone
stronger to do it!
Turning point
distance from force to turning point
force
If a greater force is applied over the same distance,
a larger moment is produced.
The moment of a force is given by:
F is the force applied, measured in Newton (N)
d is the distance from turning point, measured in metres (m)
The moment of a force is therefore measured in
Newton metres (Nm).
A force is applied to the rim of a disc which can rotate around its centre axis. In this case the moment of a force is called the Torque.
F
r
In data booklet
F is the force applied, measured in Newton (N)
r is the radius of the circle, measured in metres (m)
T is the Torque associated with force F,
measured in Newton metres (Nm).
Inertia can be defined as resistance to change in motion.
In linear motion, MASS is a measure of an object’s inertia
(since a large mass needs a large force to produce an acceleration).
In angular motion, we use MOMENT OF INERTIA.
Consider a mass m at a distance r from the axis of rotation. The moment of inertia can be calculated using:
r
In data booklet
(additional relationships)
m
m is the mass of the object, measured in kg
r is the distance from the axis of rotation, measured in m
I is the moment of inertia measured in kgm²
All the mass can be considered to be at the same distance from the axis.
Masses on a very light rod
r
Wheel with heavy rim and very light spokes
r
In these cases I=mr² where m is the total mass.
rod about centre
With:
ltotal length of the rod
m total mass of the rod
rod about end
The moment of inertia for a rodrotating about end is 4 times bigger
than the moment of inertia for a rod rotating about centre as it is
harder to do so.
This is because there are now more particles at a greater distance
from the axis of rotation.
In data booklet
(additional relationships)
r
Where m is the total mass of the disc
r is the radius of the circle
Wherem is the total mass of the sphere
r is the radius of the sphere
(all the moment of inertia formulas can be found
in the data booklet in Additional Relationships)
An unbalanced Torque will produce an angular acceleration.
In data booklet
WithI, the moment of inertia in kgm²
α, the angular acceleration in radsˉ²
T, the Torque in Nm
The angular momentum is defined as the moment of the linear momentum.
r
m
For this particle of mass m:
The linear momentum p = mv
v
w
The angular momentum = the moment of p = mvr = mr²w, since v=rw.
In data booklet
WithL angular momentum measured in kgm²sˉ¹.
A rigid body is an object in which all the individual parts have the
same angular velocity w.
The angular momentum of this body is the total of the angular
momenta of its particles:
w is constant as all particles must be rotating at the same rate.
In data booklet
In the absence of external Torque, the total angular momentum
before impact equal the total angular momentum after impact.
Not in data booklet
Before:
After:
Total angular momentum before = total angular momentum after
Iwheelw0 = w (Iwheel + Imud)
The moment of inertia of the wheel and the mud after impact is larger
than the moment of inertia of the wheel before impact.
Therefore the angular velocity of the wheel is smaller after impact.
Example: pupil spinning on a chair
After:
Before:
Pupil draws arms in
Pupil pushes arms out
Total angular momentum before = total angular momentum after
Ioutw1 = Iinw2
Iin is smaller than Iout because the particles are closer to the axis
of rotation.
Therefore w2 is larger than w1.
Example: mass dropped on a turntable
Before:
After:
Axis of rotation
Total angular momentum before = total angular momentum after
Idiscw1 = w2(Idisc+Imass)
The moment of inertia of the disc and the mass after impact is larger
than the moment of inertia of the disc before impact.
Therefore the angular velocity of the disc is smaller after impact.
In data booklet
WithI moment of inertia, measured in kgm²
w angular velocity, measured in radsˉ¹
Erot rotational kinetic energy, measured in J
Not in data booklet
WithT, Torque measured in Nm
θ, angular displacement in rad
Ew, work done in J
In the absence of frictional torque:
Not in data booklet
Ep = mgh
Erot = ½Iw²
h
w
Ek = ½mv²
v
Potential energy at top = total of linear and angular kinetic energy
mgh = ½mv² + ½Iw²
The equations of angular motion are similar to those of linear motion.
linear motion
angular motion
Example 1:
10 kg
6m
Frictional torque = 1000 Nm
time = 5s
Example 2:
r
F
Solid discmass = 20 kg
Radius = 3m
Axleradius = 1cm
Force applied to axle = 4000 N
Frictional Torque = 30 Nm
Cord length = 50 cm
Example 3: