GITIMH3 94107 Forces II Page 1

Introduction
For The Beginning Topic See Force I

Terms:
Forces Push, pull or twist which causes an object to accelerate
Inertia Tendency to resist acceleration
Momentum Tendency of an object to continue moving
   
Equilibrium A state of balance  FUp =  FDown &  FLeft =  F Right
Normal Perpendicular to
Tension Pulling force, vector quantity (N)

RF Resultant force F

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Situations
Blocks Connected by String:
m1 = 1st mass (kg)
m2 = 2nd mass (kg)
   
F
T1 F
T2 F
T3 F m3 = 3rd mass (kg)
m
1 m
2 m
3 m
4 m4 = 4th mass (kg)

FT 1 = 1st tension (N)

FT 2 = 2nd tension (N)

FT 3 = 4th tension (N)

F = Force (N)
When Force’s are Not in Equilibrium
 The force moving ‘m1’: m1.a = FT 1
 The force moving ‘m2’: m2.a = FT 2  FT 1
 The force moving ‘m3’: m3.a = FT 3  FT 2
 The force moving ‘m4’: m4.a = F  FT 3
 Force applied to pull the system: F = FT 3 + m4 a
 Acceleration: a = F/(m1 + m2 + m3 + m4)

When Force’s are in Equilibrium


 Mass’s move at a constant velocity or zero velocity (i.e. a = 0)

Two Strings Attached to a Beam Supporting a Mass:

m = Mass (kg)

1 2 FT 1 = 1st tension (N)
   
FT 1 F FT 2 FT 2 = 2nd tension (N)

FW = Weight (N)
m 
F = Normal to weight (N)
 
FW = F
FW

Force are in Equilibrium

 Weight on ‘m’: FW = m.g
 Tension in the first string: FT1 = m.g.sin1
 Tension in the second string: FT2 = m.g.sin2
 Total tension: FW = FT1.sin1 + FT2.sin2
T1.sin1 = T2.sin2

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Force Pulling a Hanging Mass to One Side:

 m = Mass (kg)

FT  FT = Tension (N)

 FW = Weight (N)
F 
m F = Force (N)


FW

Force’s are in Equilibrium

 Weight of ‘m’: FW = m.g or FW = FT.cos
 Force pulling ‘m’: F = FT.sin

 Angle between ‘ FT ’ & vertical: tan = F/FW

Blocks on a Pulley:
m1 = 1st mass (kg)
  m2 = 2nd mass (kg)
FT FT 
FT = Tension (N)

FW 1 = 1st weight (N)

m1 m2 FW 2 = 2nd weight (N)
 
FW 1 FW 2
When Force’s are Not in Equilibrium
 Weight of ‘m1’: FW1 = m1.g
 Weight of ‘m2’: FW2 = m2.g
 If m1 > m2
 
 Force moving ‘m1’ ( FW 1 > FT ): m1.a = FW1 – FT
 
 Force moving ‘m2’ ( FT > FW 2 ): m2.a = FT  FW2
 
 Resultant Force on LHS ( FW 1 > FW 2 ): (m1 + m2)a = FW1 – FW2
 
 Tension in string ( FW 1 > FT ): FW1 – FT = m1.a or FT = FW1 
m1.a
 Acceleration: a = (FW1 – FW2)/(m1 + m2)
 If m2 > m1
 
 Force moving ‘m1’ ( FT > FW 1 ): m1.a = FT  FW1
 
 Force moving ‘m2’ ( FW 2 > FT ): m2.a = FW2  FT
 
 Resultant Force on RHS ( FW 2 > FW 1 ): (m1 + m2)a = FW2 – FW1
 
 Tension in string ( FW 2 > FT ): FW2 – FT = m2.a or FT = FW2 
m2.a
 Acceleration: a = (FW2 – FW1)/(m1 + m2)

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When Force’s are in Equilibrium


 Mass’s move at a constant velocity or zero velocity (i.e. a = 0)
A Block on a Table Connected to Another Block Hanging
Vertically: 
m1 FT m1 = 1st mass (kg)

Ff m2 = 2nd mass (kg)
 
FT FW = Weight (N)

FT = Tension (N)

F f = Friction force (N)
m2


FW

When Force’s are Not in Equilibrium

 Weight: FW = m2.g
 
 Force moving ‘m1’ ( FT > F f ): m1.a = FT  Ff
 
 Force moving ‘m2’ ( FW > FT ): m2.a = FW – FT
 Tension in string: FT = m1.a + Ff or FT = FW  m2.a
 Acceleration: a = (FW – Ff)/(m1 + m2)

When Force’s are in Equilibrium

  
 Mass’s move at a constant velocity or zero velocity (i.e. a = 0 or F f > FW )

Motion on an Inclined Plane (IP):

m = Mass (kg)
  
F3 F4 FW = Weight (N)

F1 = 1st force along IP (N)
m 
 F2 = 2nd force normal down to IP (N)
F1 
  F3 = 3rd force normal up to IP (N)
FW  F2 
F4 = 4th friction force along IP (N)


When Force’s are Not in Equilibrium

 Weight of ‘m’: FW = m.g
 Force moving ‘m’ along IP: F1 = m.g.sin
 Normal down force acting on ‘m’ along IP: F2 = m.g.cos
 Acceleration along IP: a = g.sin
 Angle between horizontal and IP: tan = F1/F2

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When Force’s are in Equilibrium


 Mass’s move at a constant velocity or zero velocity (i.e. a = 0)
A Block on an IP Connected to a Block Hanging Vertically:
m1 = 1st mass (kg)
m2 = 2nd mass (kg)
m1 
 FW 1 = 1st weight (N)
F 
m2 FW 2 = 2nd weight (N)
  
FW1 FW 2 F = Force (long IP + friction) (N)

When Force’s are Not in Equilibrium

 Weight of ‘m1’: FW1 = m1.g
 Weight of ‘m2’: FW2 = m2.g
 Force ‘F’ long IP: F = Ff + m.g.sin
 If m1 > m2
 
 Force moving ‘m1’ ( F > FT ): m1.a = F – FT
 
 Force moving ‘m2’ ( FT > FW 2 ): m2.a = FT  FW2
 
 Resultant Force ( F > FW 2 ): (m1 + m2)a = F – FW2
 Tension in string: FT = m1.m2.g.sin + m1.m2.g
m1 + m2
 Acceleration: a = (F – FW2)/(m1 + m2)
 If m2 > m1
 
 Force moving ‘m1’ ( FT > F ): m1.a = FT  F
 
 Force moving ‘m2’ ( FW 2 > FT ): m2.a = FW2  FT
 
 Resultant Force ( FW 2 > F ): (m1 + m2)a = FW2 – F
 Tension in string: FT = m1.m2.g.sin + m1.m2.g
m1 + m2
 Acceleration: a = (FW2 – F)/(m1 + m2)

When Force’s are in Equilibrium


 Mass’s move at a constant velocity or zero velocity (i.e. a = 0)

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Conical Pendulum:
m = Mass (kg)

 FT = Tension (N)
 FT 
FW = Weight (N)
h l 
Fc = Centripetal force (N)
 h = Height of cone (m)
Fc l = Length of string (m)
m  
FW = Fh

FW

When Force’s are Not in Equilibrium

 Weight of ‘m’: FW = m.g
 Horizontal force up: Fh = m.g = FT.cos
 Centripetal force: Fc = m.ac = FT.sin
 Centripetal acceleration: ac = tan.g or Describing Motion II
 Period of oscillation: T = 2.. h/g
 Tension in string: FT = 2.m.l
 Angle between ‘h’ & ‘l’: cos = g/(2.l)

When Force’s are in Equilibrium


 Mass is at centre of the cone (i.e. a c = 0)

The Simple Pendulum (For Small Values of ):

m = Mass (kg)

FW = Weight (N)
 
F = Component of force on arc (N)
l l 
x = Displacement from vertical (m)
l = Length of string (m)

x m
 
F FW


When Force’s are Not in Equilibrium

 Weight of ‘m’: FW = m.g
 Acceleration: a = ( g.x)/l
 Period of oscillation: T = 2.. l/g

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When Force’s are in Equilibrium


 Mass at centre (i.e. a = 0)
Simple Harmonic Motion (SHM)
At Point(s)
 
A  Amplitude, a = Max, v = 0
 
B  Mean position, a = 0, v = Max
 
C   Amplitude, a = Max, v = 0
A AC  ½ of period

Displacement from Mean


Position

Equation: s = A . sin  .t

s = Displacement from mean position (m)

A = Amplitude (m)
 = Angular velocity (rad.s-1)
t = Time (s)

Velocity 

Equation: v =  .A . cos  .t

v = Velocity (m.s-1)

A = Amplitude (m)
 = Angular velocity (rad.s-1)
t = Time (s)

Acceleration
 
Equation: a =   2 .s

a = Acceleration (m.s-2)
 = Angular velocity (rad.s-1)

s = Displacement from mean position (m)

Hooke’s Law  
Equation: F =  k .x
F = Restoring force (N)
k = Spring constant (N.m-1)

x = Displacement of extension (m)

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Period of Oscillation
m
Equation: T = 2..
k
T = Period (s)
m = Mass (kg)
k = Spring constant (N.m-1)

Potential Energy
Equation: WPE = ½.k.s2
WPE = Potential Energy (J)
k = Spring constant (N.m-1)
s = Displacement (m)

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