Mechanics of Rigid Body. CKinematics, 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

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EQUATIONS OF MOTION OF A RIGID BODY

OO IIn 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

EQUILIBRIUM CONCEPT

Conditions for equilibrium

equilibrium = no acceleration, i.e. a particle remains at rest , or with constant velocity.

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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)

CONDITIONS FOR EQUILIBRIUM

Free Body Diagrams: Showing all the external forces acting on the body Reactions at supports and connections

Mechanics of Rigid Body. Statics

Keeping Constant Speed over an incline with angle α

Exercise: Free-Body Diagram on the rod (beam) embedded in the wall

Exercise: Free-Body Diagram on the front wheel and on the rear wheel (a) constant speed (b) accelerating (c) Acting brakes

Exercise: Free-Body Diagram on the sliding stair with friction in supporting points

Restaurant

Mechanics of Rigid Body

Sample Problem 4.1

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 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.

• 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. Statics

Mechanics of Rigid Body

Free-Body DiagramFirst 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.

• Include the dimensions necessary to compute the moments of the forces.

• 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.

• Indicate point of application, magnitude, and direction of external forces, including the rigid body weight.

Mechanics of Rigid Body. Statics

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.

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.

• 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

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.

SOLUTION:

• Create a free-body diagram for the frame and cable.

• Solve 3 equilibrium equations for the reaction force components and couple at E.

Mechanics of Rigid Body. Statics

• Create a free-body diagram for the frame and cable.

• Solve 3 equilibrium equations for the reaction force components and couple.

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Mechanics of Rigid Body. Statics

Mechanics of Rigid Body. STATICS

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2m 2.5 m

1.5m

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;

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. Restaurant

1.5 m

1 m

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.