Basic Mechanics

Md. Mohiuddin

Lecturer

Department of Mechanical Engineering

ME 2215

What is the Mechanics of Rigid Bodies?

• The mechanics of rigid bodies is subdivided into statics and dynamics.
• Statics deals with bodies at rest; dynamics deals with bodies in motion.
• In this subject, we assume bodies are perfectly rigid.
• In fact, actual structures and machines are never absolutely rigid; they deform under the loads to which they are subjected.

Fundamental Concepts and Principles?

• If the resultant force acting on a particle is zero, the particle remains at rest (if originally at rest) or moves with constant speed in a straight line (if originally in motion)

Addition of Vectors

Addition of Vectors

Problem

Two tugboats are pulling a barge. If the resultant of the forces exerted by the tugboats is a 5000-lb force directed along the axis of the barge, determine (a) the tension in each of the ropes, given that α = 45°, (b) the value of α for which the tension in rope 2 is minimum.

Component of Forces

Component of Forces- Problem

A man pulls with a force of 300 N on a rope attached to the top of a building, as shown in the figure. What are the horizontal and vertical components of the force exerted by the rope at point A?

Component of Forces- Problem

A force F = (700 lb)i + (1500 lb)j is applied to a bolt A. Determine the magnitude of the force and the angle θ it forms with the horizontal.

Component of Forces- Problem

Four forces act on bolt A as shown. Determine the resultant of the forces on the bolt.

Equilibrium of a Particle

To have equilibrium, the terminal point of the last vector must meet the initial point of the first vector.

Problem

In a ship-unloading operation, a 3500-lb automobile is supported by a cable. A worker ties a rope to the cable at A and pulls on it in order to center the automobile over its intended position on the dock. At the moment illustrated, the automobile is stationary, the angle between the cable and the vertical is 2°, and the angle between the rope and the horizontal is 30°. What are the tensions in the rope and cable?

Problem- Assignment

Determine the magnitude and direction of the smallest force F that maintains the 30-kg package shown in equilibrium. Note that the force exerted by the rollers on the package is perpendicular to the incline.

Problem

For a new sailboat, a designer wants to determine the drag force that may be expected at a given speed. To do so, she places a model of the proposed hull in a test channel and uses three cables to keep its bow on the centerline of the channel. Dynamometer readings indicate that for a given speed, the tension is 40 lb in cable AB and 60 lb in cable AE. Determine the drag force exerted on the hull and the tension in cable AC.

Problem

Two cables are tied together at C and are loaded as shown. Determine the tension (a) in cable AC, (b) in cable BC.

Problem

For W = 800 N, P = 200 N, and d = 600 mm, determine the value of h consistent with equilibrium.

Problem- Assignment

Collar A is connected as shown to a 50-lb load and can slide on a frictionless horizontal rod. Determine the magnitude of the force P required to maintain the equilibrium of the collar when (a) x = 4.5 in., (b) x = 15 in.

Problem- Assignment

A 600-lb crate is supported by several rope-and-pulley arrangements as shown. Determine for each arrangement the tension in the rope.

Rectangular Components of a Force in Space

Here F is the magnitude of the force vector

Problem

A force of 500 N forms angles of 60°, 45°, and 120°, respectively, with the x, y, and z axes. Find the components Fx, Fy, and Fz of the force and express the force in terms of unit vectors.

Rectangular Components of a Force in Space

Dividing by the magnitude F

• What is the type of vector we get when we divide a vector by its magnitude?
• What is a unit vector?

Unit vector

Rectangular Components of a Force in Space

Addition of Concurrent Forces in Space

Problem

A tower guy wire is anchored by means of a bolt at A. The tension in the wire is 2500 N. Determine (a) the components Fx, Fy, and Fz of the force acting on the bolt and (b) the angles θx, θy, and θz defining the direction of the force.

Problem

A wall section of precast concrete is temporarily held in place by the cables shown. If the tension is 840 lb in cable AB and 1200 lb in cable AC, determine the magnitude and direction of the resultant of forces exerted by cables AB and AC on stake A.

FORCES AND EQUILIBRIUM IN SPACE

A particle A is in equilibrium if the resultant of all the forces acting on A is zero.

Problem

A 200-kg cylinder is hung by means of two cables AB and AC that are attached to the top of a vertical wall. A horizontal force P perpendicular to the wall holds the cylinder in the position shown. Determine the magnitude of P and the tension in each cable

Problem

A container is supported by three cables that are attached to a ceiling as shown. Determine the weight W of the container knowing that the tension in cable AB is 6 kN.

Problem

Three cables are used to tether a balloon as shown. Knowing that the balloon exerts an 800-N vertical force at A, determine the tension in each cable.

Problem

A transmission tower is held by three guy wires attached to a pin at A and anchored by bolts at B, C, and D. If the tension in wire AB is 840 lb, determine the vertical force P exerted by the tower on the pin at A.

Problem

A container of weight W is suspended from ring A, to which cables AC and AE are attached. A force P is applied to the end F of a third cable that passes over a pulley at B and through ring A and that is attached to a support at D. Knowing that W = 1000 N, determine the magnitude of P.

Principle of Transmissibility

The principle of transmissibility states that the conditions of equilibrium or motion of a rigid body remains unchanged if a force F acting at a given point of the rigid body is replaced by a force F’ of the same magnitude and the same direction, but acting at a different point, provided that the two forces have the same line of action.

Moment of a Force about a Point

d represents the perpendicular distance from O to the line of action of F

The magnitude of M_O measures the tendency of the force F to make the rigid body rotate about a fixed axis directed along M_O.

Two Dimensional Problems

Rigid-Body Equilibrium in Two Dimensions

Reactions in two Dimensions

Problem

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.

Problem

A loading car is at rest on a track forming an angle of 25° with the vertical. The gross weight of the car and its load is 5500 lb, and it acts at a point 30 in. from the track, halfway between the two axles. The car is held by a cable attached 24 in. from the track. Determine the tension in the cable and the reaction at each pair of wheels.

Problem

The frame shown supports part of the roof of a small building. Knowing that the tension in the cable is 150 kN, determine the reaction at the fixed end E.

Practice Problem

A load of lumber of weight W = 25 kN is being raised by a mobile crane. The weight of boom ABC and the combined weight of the truck and driver are as shown. Determine the reaction at each of the two (a) front wheels H, (b) rear wheels K.

Practice Problem

Determine the reactions at A and C when (a) α = 0, (b) α = 30°.

Problem

A slender rod AB with a weight of W is attached to blocks A and B that move freely in the guides shown. The blocks are connected by an elastic cord that passes over a pulley at C. (a) Express the tension in the cord in terms of W and θ. (b) Determine the value of θ for which the tension in the cord is equal to 3W.

Problem

An 8-kg slender rod of length L is attached to collars that can slide freely along the guides shown. Knowing that the rod is in equilibrium and that β = 30°, determine (a) the angle θ that the rod forms with the vertical, (b) the reactions at A and B.

Problem

The smooth disks D and E have a weight of 200 Ib and 100 lb, respectively. Determine the largest horizontal force P that can be applied to the center of disk E without causing disk D to move up the incline.

Practice- Problem

One end of rod AB rests in the corner A and the other end is attached to cord BD. If the rod supports a 150-N load at its midpoint C, find the reaction at A and the tension in the cord.

Practice- Problem

For the frame and loading shown, determine the reactions at C and D

Rigid-Body Equilibrium in Three Dimensions

Some equilibrium problems might involve individual couples M_i either as applied loads or as support reactions. In such situations, you can accommodate these couples by expressing the second part

Rigid-Body Equilibrium in Three Dimensions

Reactions in three Dimensions

Reactions in three Dimensions

In these two cases, couple moments are not applied if the body is supported elsewhere.

Problem

A uniform pipe cover of radius r = 240 mm and mass 30 kg is held in a horizontal position by the cable CD. Assuming that the bearing at B does not exert any axial thrust, determine the tension in the cable and the reactions at A and B.

Problem

The rigid L-shaped member ABC is supported by a ball-and-socket joint at A and by three cables. If a 1.8-kN load is applied at F, determine the tension in each cable.

Moment of a Force about a Given Axis

Though the moment produced at point O due to force F has x and y components, only the y components of moment is necessary as this moment helps to rotate the nut.

Moment of a Force about a Given Axis

Problem

Determine the moment M_AB, produced by the force F which tends to rotate the rod about AB axis. Also find the magnitude.

Practice Problem

Determine the moment M_OA, produced by the force F.

Moment of a Couple

Two forces F and -F, having the same magnitude, parallel lines of actions but acting in opposite direction, are said to form a couple.

The sum of the moments of the two forces about O,

r is extended from any point on the line of action of force –F to any point on the line of action of force F

Moment of a Couple

The vector M is called the moment of the couple

Magnitude

You can consider a couple as a moment

Problem

Determine the resultant couple moment of the three couples acting on the plate

Problem

Determine the couple moment acting on the pipe

Simplification of Force Couple System

Any force F acting on a rigid body can be moved to an arbitrary point O provided that we add a couple whose moment is equal to the moment of F about O.

Problem

Replace the force and couple system shown in the figure with an equivalent resultant force and couple moment acting at point O.

COG of a Two-Dimensional Body

Centroids of Areas

• The centroid is the geometric center.
• If the body is made of homogeneous material centroid and center of gravity coincide.

Centroids of lines

COGs and Centroids of Areas and Lines

COGs and Centroids of Areas and Lines

COGs and Centroids of Areas and Lines

COGs and Centroids of Areas and Lines

COGs and Centroids of Areas and Lines

Using

COGs and Centroids of Areas and Lines

Similarly, for line,

Using

Problem 1

For the plane area shown, determine (a) the first moments with respect to the x and y axes; (b) the location of the centroid.

Problem 1 (Solution)

Problem 2

Determine the location of the centroid.

Practice Problem

Practice Problem

Practice Problem

Practice Problem

Centroids of Areas by Integration

Problem 1

Determine the location of the centroid of a parabolic spandrel by direct integration.

Practice Problems

Practice Problems

Centroid of Volume

By doing a similar analysis as was done for the area, it can be shown that

Thank You