Mechanics of Rigid Body. C
Kinematics, Kinetics and Statics
1.- Introduction
2.- Kinematics. Types of Rigid Body Motion:
Translation,
Rotation
General Plane Motion
3.- Kinetics. Forces and Accelerations. Energy and
Momentum Methods.
Angular Momentum and Moment of Inertia
Fundamental Equations of Dynamics
4.- Statics. Equilibrium.
Mechanics of Rigid Body
EQUILIBRIUM CONCEPT
Conditions for equilibrium
equilibrium = no acceleration, i.e. a particle remains at rest , or with
constant velocity.
EQUATIONS OF MOTION OF A RIGID BODY
 Fext  m aCM
 CM  ICM 
 O  IO
In the case of a rotation about a fixed axis and
O is a point of this axis. In general, if O is a
point of a inertial reference system
CONDITIONS FOR EQUILIBRIUM
 Fext  O
  0
1.- The net external force acting on the
body must remain zero.
2.- The net external torque about any
point must remain zero. Any point can be
considered because all of them are in a inertial
reference system (non accelerated)
Mechanics of Rigid Body. Statics
Free Body Diagrams:
Showing all the external forces acting on
the body Reactions at supports and
connections
Restaurant
Exercise: Free-Body Diagram on the
front wheel and on the rear wheel (a)
Exercise: Free-Body Diagram constant speed (b) accelerating (c)
on the rod (beam) embedded Acting brakes
in the wall
Exercise: Free-Body Diagram
on the sliding stair with
friction in supporting points
Keeping Constant Speed over
an incline with angle α
Mechanics of Rigid Body.
Body Statics
Sample Problem 4.1
• Determine B by solving the equation for
the sum of the moments of all forces
about A. Note there will be no
contribution from the unknown
reactions at A.
A fixed crane has a mass of 1000 kg
and is used to lift a 2400 kg crate. It
is held in place by a pin at A and a
rocker at B. The center of gravity of
the crane is located at G.
Determine the components of the
reactions at A and B.
• Determine the reactions at A by
solving the equations for the sum of
all horizontal force components and
all vertical force components.
• Check the values obtained for the
reactions by verifying that the sum of
the moments about B of all forces is
zero.
Mechanics of Rigid Body.
Body Statics
Free-Body Diagram
First step in the static equilibrium analysis of a rigid
body is identification of all forces acting on the
body with a free-body diagram.
• Select the extent of the free-body and detach it
from the ground and all other bodies.
• Indicate point of application, magnitude, and
direction of external forces, including the rigid
body weight.
• Indicate point of application and assumed
direction of unknown applied forces. These
usually consist of reactions through which the
ground and other bodies oppose the possible
motion of the rigid body.
• Include the dimensions necessary to compute
the moments of the forces.
Mechanics of Rigid Body. Statics
SOLUTION:
• Create a free-body diagram of the joist.
Note that the joist is a 3 force body acted
upon by the rope, its weight, and the
reaction at A.
A man raises a 10 kg joist, of
length 4 m, by pulling on a rope.
Find the tension in the rope and
the reaction at A.
• The three forces must be concurrent for
static equilibrium. Therefore, the reaction
R must pass through the intersection of the
lines of action of the weight and rope
forces. Determine the direction of the
reaction force R.
Mechanics of Rigid Body. Statics
SOLUTION:
• Create a free-body diagram for the
frame and cable.
• Solve 3 equilibrium equations for the
reaction force components and
couple at E.
The frame supports part of the roof of
a small building. The tension in the
cable is 150 kN.
Determine the reaction at the fixed
end E.
Mechanics of Rigid Body. Statics
• Solve 3 equilibrium equations for the
reaction force components and couple.
4.5
150 kN   0
F

0
:
E

 x
x
7.5
E x  90.0 kN
 Fy  0 : E y  420 kN  
6
150 kN   0
7.5
E y  200 kN
• Create a free-body diagram for
the frame and cable.
 M E  0 :  20 kN 7.2 m   20 kN 5.4 m 
 20 kN 3.6 m   20 kN 1.8 m 

6
150 kN 4.5 m  M E  0
7.5
M E  180.0 kN  m
Mechanics of Rigid Body. STATICS
A uniform plank of length L = 3 m and mass
M=35 kg is supported by scales a distance d
=0.5 m from the ends of the board, as shown
in the figure. A block B of mass mB=50 kg is
placed on the board, as indicates in the figure.
Determine the reactions on the supports.
The sign of the restaurant has a mass of 30 kg and
hangs from a rod of mass 100 kg and length of 3 m,
pivoted at the wall and supported by a cable as
shown in the figure. Calculate (a) the tension on the
cable (b) the reactions on the pivot.
1m
1.5 m
Restaurant
Several men pulling on a cable maintain the truck at rest, as shown in the figure.
Neither brakes or motor are acting, then (a) calculate the tension in the cable, (b)
the reactions on the front and on the rear wheels. Data: Aspect: 10º; mass of the
truck: 5.000 kg: height of the cable from the soil : 1 m;
CM
1.5m
2m
2.5 m