Examples Of Central Forces

1. If the force on a body is always
towards a fixed point, it is called a
central force. Take the fixed point
as the origin.
Examples of central forces
1. uniform circular motion
2. force due to gravitation
3. simple harmonic motion
4. projectile motion
5. uniformly accelerated motion
6. others, like electrostatic , magnetostatic
forces, etc.

2.  By studying central forces you may
master
 1. uniform circular motion
2. force due to gravitation
3. simple harmonic motion
4. projectile motion
5. uniformly accelerated motion
at the same time !
All

3. Since forces involve mass and
acceleration, acceleration involves
differentiation of velocity, velocity is
differentiation of displacement, we need to know
differentiation prior to it.
Since displacement, velocity, acceleration and
force are vector quantities, we need to know
vectors prior to it. Then what we are required to
know is vectors, differentiation and vector
differentiation of course.

4. differentiation of vector functions of scalar
variable- time in Cartesian coordinates
vector r of a moving mass point may be
(Position

resolved into x and y components in Cartesian
coordinates as r cos and r sin respectively. We write
r = x + y = r cos i + r sin j ……………………….(1)

where i and j are unit vectors in x and y directions

respectively.
On differentiation, we get,

or , v = vx + vy………………………...………….(2)

where vx and vy as respectively and velocity is vector

differentiation of position vector.

5. DIFFERENTIATION OF VECTORS CARTESIAN COORDINATES
(CONTINUED FROM PREVIOUS SLIDE)
 where vx and vy as respectively and velocity is
vector differentiation of position vector.
 Eqn.(2) makes an important statement that the
components of velocity in Cartesian
coordinates are time derivatives of the
components of position vectors. This result
appears too obvious, but as we would see later, it
may not hold in other system of coordinates .A
second differentiation gives
 or , a = ax + ay………………………….….(3)

6. DIFFERENTIATION OF VECTORS CARTESIAN COORDINATES
(CONTINUED FROM PREVIOUS SLIDE)
dv y
dv x
where ax and ay are dt a n d dt respectively or
2 2
respectively as y
dx d
and
2 2
acceleration t vector differentiation of
dt d
is
velocity vector.
Eqn.(3)similarly states that the components of
acceleration in Cartesian coordinates are
time derivatives of the components of
velocity vectors. Again it may not hold in
other system of coordinates.

7. DIFFERENTIATION OF VECTORS POLAR
COORDINATES
Y
Q
s
r
P
r+ r
Y
s /2+
r y
T
r
X
O x X
R
Fig 1:Resolution of radius vector into
components

8. DIFFERENTIATION OF VECTORS POLAR
COORDINATES
Instead of differentiating displacement and velocity

vectors, let us differentiate unit vectors and θ
r
(taken ┴ to each other) . Expressing them in Cartesian
coordinates, or resolving into components
i + sin j and θ= - sin i + cos j ….(5)
 r =cos
 Since magnitudes of both of them unity but directions are
both variables . (see the figure in the above slide, no 7.
 For differentiation of the unit vectors refer to the figure in
the next slide. Later on the formula for differentiation of
unit vectors shall be fruitfully utilised for differentiating
displacement and velocity vectors.

9. The unit vectors , , their increments r r
r
,are shown in the figure.
Q S
Q
P
r r
r r
r T
P
r
A’ A O P
r=1 O S
x
Fig 2 : differentiation of unit vectors

10. DIFFERENTIATION OF UNIT VECTORS.
as the unit vector makes an angle with the x – axis and the unit vector

makes an angle /2+ with the x – axis and both the unit vectors have
obviously magnitudes unity. Mind it that and are unit vectors θ
r
continuously changing in direction and are not constant vectors as such;
whereas i and j are constant vectors.
 Differentiating the unit vectors with respect to time t, we have,(from
(5) above) d r sin d i cos d j and d θ cos d i sin d j respectively
dt dt dt dt
dt dt
or, d r dθ
and respectively,
 d d d d
i cos j θ cos i j r
sin sin
dt dt
dt dt dt dt
dr
dθ
and respectively…………………..……….(6)
θ
or
 r
dt
dt
d
where , the magnitude of angular velocity of the moving particle
 dt
around the point O, or the time rate of turning of .
dr
It is important to see here that is parallel to , i.e.,
 θ
dt
perpendicular to , ri.e., in a direction tangent to the unit circle. Also

d
is parallel to , i.e., along the radius and towards the
r
dt dr
2
θ Thus
center, and thus it is perpendicular to . is parallel
 2
dt
to d , i.e., parallel to r.

dt
Thus the derivative of is in the direction of orr centripetal.
 θ

11. DIFFERENTIATION OF VELOCITY AND ACCELERATION VECTORS

12. WHAT IF THE FORCE IS ALWAYS TOWARDS A FIXED POINT, I.E., CENTRAL FORCE

13. Different cases of central force
.. . .
r θ
2
mr r m 2r r
F = ma, then Frrˆ + Fθ =
θ
..
1. For uniform circular motion, r =a, ω is a constant and r 0
since r is a constant. So F rˆ = - a 2 Fθ=0 ..
2. For simple harmonic motion, Fθ=0, ω =0, r kr
3. For projectile motion, simpler will be Cartesian coordinates, ax =0,
and ay =-g, and uniform acceleration is a particular case of
projectile motion where the horizontal velocity is 0 always.