Circular Motion,Relative Motion And SHM Flashcards
Circular motion definition
When a particle moves in a plane such that it’s distance from a fixed(or moving) point remains constant, then it is said to be in circular motion with respect to that point.
Position vector definition
A vector that establishes the position of a point in space relative to the arbitrary reference origin.
Angular position of a point in space requires
An arbitrary origin and a reference line.
Angular position of a point in space is defined as
The angle made by the position vector(w.r.t origin) with the reference line.
Circular motion has …. dimensions
2
2 types of acceleration in circular motion
Tangential acceleration
Centripetal acceleration
Tangential acceleration direction
Tangential to the circular motion.
Tangential acceleration changes the
Speed of the particle.
Centripetal acceleration changes the
Direction of velocity
Centripetal acceleration is also called
Radial acceleration or normal acceleration (R).
Total acceleration (in circular motion)
Vector sum of tangential and centripetal acceleration (R).
In circular motion there must be a
Force acting on the object otherwise, it should follow a linear motion.
Motion definition
Change in position of an object over time.
W
Relative position
Position of a particle with respect to the observer.
R
The position of A w.r.t B is denoted by the
Displacement vector.
Position vector of A - Position vector of B.
And vice versa.
Relative velocity of A with respect to B is
The velocity with which A appears to move, if B is at rest. (R)
Periodic motion definition
When body repeats its motion along a definite path in regular intervals of time, it’s motion is said to be Periodic motion. (R)
The regular intervals of time in which the periodic motion of an object repeats it
Time period or
Harmonic motion period (R)
Time period definition
Smallest time interval after which the oscillation repeats itself. (R)
Time period of a second pendulum
2 seconds. (R)
Length of a second pendulum
.993 m.
Angular displacement definition
Angle through which the position vector of a moving particle rotates in a given interval of time. (R)
Angular displacement depends on
The origin ,but does not depend on the reference line. (R)
Angle, Angular displacement, Angular velocity, Angular acceleration are all
Dimensionless quantities.
1 rev =
360°
1 revolution written as
1 rev
Angular velocity is represented by
Omega
Angular acceleration is represented by
Alpha
Axial vectors definition
Vectors representing rotation effect. (R)
If angular acceleration is 0 ,then circular motion is
Uniform.
Angular acceleration acts along the
Axis of rotation
Tangential acceleration acts
Tangentially to the circular path
Tangential acceleration =
radius x angular acceleration
Centripetal acceleration =
V*2/r
Velocity of approach of 2 moving particles =
|V(ab)| = |V(ba)|
V(ab)
Velocity of a with the respect to b, where b itself is considered to be at rest.
Oscillatory motion
To and fro type of motion of objects
Damped oscillations
Oscillations in which energy is consumed due to some resistive forces, thus causing the total mechanical energy to reduce.
Restoring force
Force acting in an oscillation directed towards the equilibrium point.
Simplest form of oscillatory motion
Simple Harmonic Motion
Definition of SHM
If the restoring force acting on the body is directly proportional to the displacement of the body and is always directed towards the equilibrium position.
Types of SHM
Linear SHM
Angular SHM
In a linear SHM, the oscillation happens along a
Straight line.
In an oscillation there is
2 extreme positions and 1 mean position(equilibrium point)
Amplitude of a particle in SHM
Maximum value of displacement of a particle from its equilibrium position.
Amplitude of a particle in SHM motion is directly proportional to the
Total mechanical energy of a body.
Amplitude of a particle in SHM is quantitatively expressed as
A = 1/2 x (distance between the 2 extreme positions)
For SHM to exist, only 1 condition is sufficient ?
F = -kx
x : displacement
k : spring constant
Spring constant is also called as
Force constant.
The linear displacement between 2 points in circular motion is
r∆θ
s = rθ , prove it ?
S=(theta/2pi)(2pi*r)=r(theta)
Proof of v = r ω
v - linear speed
r - radius
ω - angular velocity
∆s = r∆θ ∆s/∆t = r∆θ/∆t v = r ω
Relationship btw angular velocity and linear velocity
v = r ω
a(t) = r α
Derivation ?
By differentiating the equation v = r ω with respect to time.
Tangential acceleration is the
Rate of change of speed and not velocity.
In v = r ω,
v is
Linear speed and not linear velocity.
Linear velocity vector is always
Tangential
Linear speed =
Radius x angular velocity
v=r α
Linear velocity
r ω [-i sinθ + j cosθ] = r ω e(t)
Linear acceleration =
-ω²r e(r) + dv/dt e(t)
e(r) and e(t) are unit vectors, with specific directions.
e(r) : radial direction
e(t) : tangential acceleration
e(r) is the
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Radial unit vector = [i cosθ + j sinθ]
e(t) is the
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Tangential unit vector = [-i sinθ + j cosθ]
If it is uniform circular motion, then ….. is not present
Tangential acceleration
Tangential acceleration =
dv/dt
Resultant acceleration makes an angle alpha with the radius , and tan alpha =
(dv/dt)/(v²/r)
Derivation of a(c) = v²/r from a(c) = ω²r
ω²r = (v²/r²) x r = v²/r
Angular acceleration is a vector , but angular velocity is a
Scalar
Angular displacement is
Scalar
Angular displacement is a
Scalar, unless it is infinitesimally small.
Distance travelled in n second
sn=u+a/2(2n-1)
If the velocity of an object is less than √5gl then the motion of the particle depends on whether
The speed or tension will become 0 first.
Conditions for speed to become 0 before tension
0 < velocity < √2gl
This happens only between 0° and 90°
If the velocity is √2gl, then
The particle travels 90° and speed and tension becomes 0 at the same time.
If speed becomes 0 before tension, then the object will
Oscillates
Conditions for tension to become 0 first
√2gl < velocity < √5gl
This happens only between 90° and 180°
If the object’s tension becomes 0 first, then the motion would be
A combined form of circular and projectile motion.
Newton’s law are not valid in a
Non inertial frame.
Centrifugal force is a
Pseudo force
Pseudo forces have to be assumed if the frame in which we are working is
Non inertial
Centrifugal force =
Centripetal Force = mv2/r
If we analyse the motion of objects moving in a rotating frame many pseudo forces have to be assumed like
Coriolis Force,centrifugal Force etc.
If we are working in an inertial frame
No pseudo forces have to be applied.
Earth’s axis of rotation
Line joining North Pole and South Pole.
Every point in earth
Moves in a circle
Poles of the earth
Do not rotate and hence effect of earth’s rotation is not felt there.
Relationship between mg and mg’ in poles and equator
In poles
mg = mg’
In equator
mg’ - mv*2/r = mg
g’ ≥ g
T
g’ =
g + mv²/r = √[g² − ω²Rsin²θ (2g-ω²r)]
r =
R sin θ
To convert Radians into degrees multiply it by
180/pie = 57.3
Cos θ for very small angles (in Radians) can be found by
1-(θ²/2)
