Mechanics 2 Flashcards
What is the axis of rotation?
The centers of the circles traced by the rotating particles lie on a straight line called the axis of rotation which is fixed and perpendicular to the planes of the circles.
Define rigid body
A rigid body is one whose geometric shape and size remain unchanged under the action of any external force.
Define moment of a force or torque
The ability of a force to produce rotational motion is called moment of a force or torque.
What kind of motion does a couple produce? Why?
Rotational only, not translational.
Net force acting on the rigid body is zero.
What is rotational inertia of a body?
The quantity that measures the inertia of rotational motion of the body is called the rotational inertia or moment of inertia of the body.
Expression for kinetic energy of a rigid body rotating with constant angular speed
E= (1/2) Iω*ω
Define moment of inertia
The moment of inertia of a rigid body about an axis of rotation is defined as the sum of the product of the mass of each particle and the square of its perpendicular distance from the axis of rotation.
What factors influence moment of inertia of a body?
Mass, shape and size of the body
Distribution of mass of the body about the axis of rotation
Position and orientation of the axis of rotation
Give the physical significance of I
- F=ma and T=Ia
2. Et=(1/2)mvv and Er=(1/2)Iωω
Similarities and difference between mass and moment of inertia
Mass and MI remain constant.
But unlike mass, MI depends on position of axis of rotation, shape and size of the body, and distribution of mass about the axis of rotation.
How are flywheels and grinding wheels designed?
Flywheel: more mass distributed near the rim that at centre region near the axis. Also large diameter. So, MI increases.
Grinding wheel : large mass, moderate diameter. So, MI large, maintains motion and does not stop quickly when set into motion.
Define Radius of Gyration
If the whole mass of body ‘M’ is at a radial distance ‘K’ from the axis of rotation then the moment of inertia remains same about that axis of rotation, then ‘K’ is called radius of gyration of the body.
Radius if gyration - Equation
I = MKK
Physical significance of radius of gyration
Depends on shape and size of the body.
Measures distribution of mass about the axis of rotation.
Small value- mass distributed closer to axis of rotation and MI small
Large value- mass distributed at large distance from axis of rotation so MI large
Direction of torque and angular acceleration
Directed Parallel to the axis of rotation of the body
Equation for total KE for rolling motion
E= (1/2) Mvv[1+ (KK/(R*R))]
Equation for velocity of rolling motion down an inclined plane
V= root( 2gh/(1+(KK/(RR))))
Equation for acceleration of rolling motion down and inclined plane
a= gsin(θ) / (1+(KK/(RR)))
State the theorem of parallel axes
The theorem of perpendicular axes states that the moment of inertia of a body about any axis is equal to the sum of the it’s moment of inertia about a parallel axis passing through its centre of mass and the product of its mass and the square of the perpendicular distance between the two perpendicular axes.
State the theorem of perpendicular axes
The theorem of perpendicular axes states that the Moment of Inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of its moment of Inertia about two mutually perpendicular axes concurrent with the perpendicular axis and lying in the plane of the laminar body.
Limitation of theorem of perpendicular axes
Only for laminar bodies
MI about a thin uniform rod about an axis passing through its centre of mass and perpendicular to its length
I = MLL/12
MI of a thin ring about a transverse axis passing through its centre
I=MRR
Moment of inertia of a ring about its diameter
I=MRR/2
MI of a ring about a tangent in its plane
I=3MRR/2
MI of a ring about a tangent perpendicular to its plane
I=2MRR
MI of a disc/solid cylinder about an axis passing through its centre and perpendicular to its plane
I=MRR/2
MI of a solid cylinder about transverse axis passing though it’s centre
I= MRR/4 + MLL/12
MI of a uniform solid sphere about a diameter
I= 2MRR/5
Equation for angular momentum
Angular momentum vector= radius vector x linear momentum vector
L=Prsin(θ)
L=mvrsin(θ)
L=Iω
Direction of Angular Momentum
Same as that of ω vector along axis of rotation
State the principle of conservation of angular momentum
It states that the angular momentum of a rotating body is conserved or remains constant if the resultant external torque acting on the body is zero.
Examples of conservation of angular momentum
extend out =large MI change
Diver executing a somersault
Ballet dancers and ice skaters increase speed of rotation
Person standing on rotating chair
What does the term periodic indicate?
Motion, processes and phenomenon which repeats in equal intervals of time is called periodic.
What is periodic motion?
Motion that repeats itself in equal intervals of time is called periodic motion.
What is vibrating periodic motion?
In vibratory motion when the particle is repeating its motion, to and fro, along the same path in equal intervals of time is vibrating periodic motion.
Equation for restoring force
F=-kx
Define linear SHM
Linear SHM is defined as the linear periodic motion of a body in which the restoring force (or acceleration) is always directed towards the mean position and it’s magnitude is directly proportional to the displacement from the mean position.
Differential equations of SHM
d2x/dt2 + (ω^2)x= 0
Pg 57
Equation for velocity of a particle performing SHM
v= +/- (ωroot((aa)-(xx)))
Expression for displacement of a particle performing SHM
x=asin(ωt+α)
Maximum displacement of SHM
Extreme position
Minimum displacement of SHM
Mean position
Maximum velocity of SHM
Mean position
Minimum velocity of SHM
Extreme position
Maximum acceleration of SHM
Extreme position
Minimum acceleration of SHM
Mean position
Amplitude of a particle performing SHM
The magnitude of maximum displacement of the particle performing SHM from its mean position or equilibrium position is called amplitude of SHM.
Define oscillation
In SHM the particle performs the same set of movements again and again. Such one set of movements is called and oscillation (or vibration).
Define Period of SHM
In SHM the time required for the particle to complete one oscillation is called period of SHM.
Give the expression for Time period of SHM
T= 2π / root(acceleration per unit displacement)
How does the period of SHM depend on amplitude, energy and phase constant?
Independent
Define frequency of SHM
The number of oscillations performed per unit time by a particle performing SHM is called frequency of SHM.
Equation for frequency of SHM
f= (1/2π) root(k/m)
Define phase of SHM
The physical quantity that describes the state of oscillation, that is, the magnitude and direction of displacement of particle at the given instant is called phase of SHM.
Define epoch of SHM
The physical quantity which describes the state of oscillation of particle performing SHM at the start of motion is called epoch of SHM.
Kinetic energy of a particle performing SHM
KE= (1/2)k((aa)-(xx))
Potential energy of a particle performing SHM
PE = (1/2) k(x*x)
Total energy of a particle performing SHM
TE= (1/2)k(a*a) = (1/2)mωωaa
What does the total energy of a particle performing SHM depend on?
- Directly proportional to its mass
- Directly proportional to the square of frequency of oscillation
- Directly proportional to the square of amplitude of oscillation
When is TE=KE? (SHM)
Mean position
When is TE=PE? (SHM)
Extreme position
Describe the nature of graph of KE, PE against displacement. (SHM)
Parabolic
Resultant amplitude of combination of two SHMs
RR = a1a1 + a2a2 + 2a1a2cos( α1-α2)
Resultant phase of combination of two SHMs
δ= tan inverse( (a1sin(α1)+a2sin(α2))/ (a1cos(α1)+a2cos(α2)))
What does the initial phase angle, delta, of a combination of two SHMs depend on?
- Initial phase angles α1 and α2
2. Amplitudes a1 and a2
Define ideal simple pendulum
And ideal simple pendulum is defined as a heavy particle (point mass) suspended by a weightless, inextensible and twist less string from a perfectly rigid support.
Define a practical pendulum
A practical pendulum is defines as a heavy sphere (bob) suspended by a light inextensible string from a rigid support.
Define length of a simple pendulum.
Length of a simple pendulum is defined as distance between point of suspension and centre of gravity of the heavy sphere.
Restoring force on a simple pendulum
F= -mgsin(θ)
Restoring force for a simple pendulum performing SHM
F= -mgx/L
Acceleration of a simple pendulum performing SHM
Acceleration= -gx/L
Time period of a simple pendulum performing SHM
T= 2π root(L/g)
Frequency of a simple pendulum performing SHM
f= (1/2π) root(L/g)
What is a seconds pendulum?
A simple pendulum whose period is 2 seconds is called a seconds pendulum.
Expression for length of a seconds pendulum
L= g/(π*π)
Laws of simple pendulum include:
- Law of length
- Law if acceleration due to gravity
- Law of mass
- Law of isochronous
State the law of length of simple pendulum
The period of a pendulum is directly proportional to the square root of its length.
State the law of acceleration due to gravity of a simple pendulum.
The period of a pendulum is inversely proportional to the square root of acceleration due to gravity.
State the law of mass of a simple pendulum
The period of a simple pendulum does not depend on its mass.
State the law of isochronous of a simple pendulum
The period of simple pendulum does not depend on its amplitude.
When is an oscillator said to be damped?
When the motion of and oscillator is reduced by and external force, the oscillator and it’s motion are said to be damped.
What are damped harmonic oscillations and what is a damped harmonic oscillator?
Periodic oscillations of gradually decreasing amplitude are called damped harmonic oscillations and the oscillator is called a damped harmonic oscillator.
In and idealized example of a block attached to a vane submerged in water, what does the dampening force depend on?
- Nature of surrounding medium
2. Directly proportional to velocity of the vane and block
Differential equation of damped oscillations
m* (d2x/dt2) + b(dx/dt) +kx=0
Amplitude of damped harmonic oscillations
Ae^(-bt/2m)
Displacement of particle performing damped SHM
x=A(e^(-bt/2m))cos(ω’t+(φ))
Angular frequency of particle performing damped oscillations
ω’ = root((k/m)-((b/2m)^2))
Time period of oscillations of a particle performing damped oscillations
T = 2π/( root((k/m)-((b/2m)^2)))
What happens to time period and amplitude because of the dampening force?
Period increases
Amplitude decreases
