Formula Sheets: Kinematics of Point Mass & Rigid Bodies

 Page 1


GA TE CE 2026 F orm ula Sheet: F riction, Cen tre of Mass, and
F r ee V ibrations
1. F rictions and its Applications
• F rictional force (static or kinetic):
F
f
=µ ·N
where:
– F
f
: F rictional force (N)
– µ : Co e?icien t of friction (static µ
s
or kinetic µ
k
)
– N : Normal force (N)
• Maxim um static frictional force:
F
f, max
=µ
s
·N
• Angle of friction:
? = tan
-1
(µ
s
)
where:
– ? : Angle of friction (degrees)
• Angle of rep ose:
? = tan
-1
(µ
s
)
where:
– ? : Angle of rep os e (degrees)
• F orce required to initiate motion on an inclined plane:
F =W ·(sin? +µ
s
·cos?)
where:
– F : F orce parallel to incline (N)
– W : W eigh t of ob ject (N)
– ? : Angle of incline (degree s)
• F orce to main tain motion (kinetic friction):
F
k
=W ·(sin? +µ
k
·cos?)
2. Cen tre of Mass
• Cen tre of mass co ordinates (discrete masses, 2D):
x
c
=
?
m
i
x
i
?
m
i
, y
c
=
?
m
i
y
i
?
m
i
where:
– x
c
, y
c
: Co ordinates of cen tre of mass (m)
– m
i
: Mass of particle i (kg)
– x
i
, y
i
: Co ordinates of particle i (m)
• Cen tre of mass for con tin uous b o dy (2D):
x
c
=
?
xdm
?
dm
, y
c
=
?
ydm
?
dm
where:
1
Page 2


GA TE CE 2026 F orm ula Sheet: F riction, Cen tre of Mass, and
F r ee V ibrations
1. F rictions and its Applications
• F rictional force (static or kinetic):
F
f
=µ ·N
where:
– F
f
: F rictional force (N)
– µ : Co e?icien t of friction (static µ
s
or kinetic µ
k
)
– N : Normal force (N)
• Maxim um static frictional force:
F
f, max
=µ
s
·N
• Angle of friction:
? = tan
-1
(µ
s
)
where:
– ? : Angle of friction (degrees)
• Angle of rep ose:
? = tan
-1
(µ
s
)
where:
– ? : Angle of rep os e (degrees)
• F orce required to initiate motion on an inclined plane:
F =W ·(sin? +µ
s
·cos?)
where:
– F : F orce parallel to incline (N)
– W : W eigh t of ob ject (N)
– ? : Angle of incline (degree s)
• F orce to main tain motion (kinetic friction):
F
k
=W ·(sin? +µ
k
·cos?)
2. Cen tre of Mass
• Cen tre of mass co ordinates (discrete masses, 2D):
x
c
=
?
m
i
x
i
?
m
i
, y
c
=
?
m
i
y
i
?
m
i
where:
– x
c
, y
c
: Co ordinates of cen tre of mass (m)
– m
i
: Mass of particle i (kg)
– x
i
, y
i
: Co ordinates of particle i (m)
• Cen tre of mass for con tin uous b o dy (2D):
x
c
=
?
xdm
?
dm
, y
c
=
?
ydm
?
dm
where:
1
– dm : Differen tial mass elemen t (kg)
• Cen tre of mass for comp osite b o dies:
x
c
=
?
A
i
x
i
?
A
i
, y
c
=
?
A
i
y
i
?
A
i
where:
– A
i
: Area of section i (m²)
– x
i
, y
i
: Co ordinates of cen troid of section i (m)
3. F ree Vibrations of Undamp ed SDOF System
• Equation of motion (undamp ed SDOF):
m¨ x+kx = 0
where:
– m : Mass (kg)
– ¨ x : A cceleration (m/s²)
– k : Spring stiffness (N/m)
– x : Displacemen t (m)
• Natural frequency:
?
n
=
v
k
m
where:
2