Statics of Particles and Vector lecture no-2 - Engineering Mechanics
Subject: Engineering Mechanics
Determination of direction of minimum T2 .
Triangle rule applied for minimum T2.
Rectangular Components of a Force: Unit Vectors
Fx and Fy are referred to as the scalar components of F
Subject Code: ME 4301
Topic: Statics of Particles and Vector lecture no-2
Lecturer: Md. Rezaul Karim
•The objective for the current chapter is to investigate the effects of forces on particles:
-replacing multiple forces acting on a particle with a single equivalent or resultant force,
-relations between forces acting on a particle that is in a state of equilibrium.
•The focus on particles does not imply a restriction to miniscule bodies. Rather, the study is restricted to analyses in which the size and shape of the bodies is not significant so that all forces may be assumed to be applied at a single point.
Resultant of Two Forces
•force: action of one body on another; characterized by its point of application, magnitude, line of action, and sense.
•Experimental evidence shows that the combined effect of two forces may be represented by a single resultant force.
•The resultant is equivalent to the diagonal of a parallelogram which contains the two forces in adjacent legs.
•Force is a vector quantity.
•Scalar: parameters possessing magnitude but not direction. Examples: mass, volume, temperature
•Vector classifications:
-Fixed or bound vectors have well defined points of application that cannot be changed without affecting an analysis.
-Free vectors may be freely moved in space without changing their effect on an analysis.
-Sliding vectors may be applied anywhere along their line of action without affecting an analysis.
-Equal vector shave the same magnitude and direction.
-Negative vector of a given vector has the same magnitude and the opposite direction.
Parallelogram law of Forces
If two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is presented in magnitude and direction by the diagonal passing through the point.
Sample Problem:
A barge is pulled by two tugboats. If the resultant of the forces exerted by the tugboats is 5000 lbf directed along the axis of the barge, determine
a)the tension in each of the ropes for a= 45 degree,
b)the value of a for which the tension in rope 2 is a minimum.
Solution:
a. Tension for a=45 degree.
Graphical Solution: use the parallelogram law. The resultant the diagonal of the parallelogram is equal to 5000 lb and is directed to the right. Draw the sides parallel to the ropes. If the drawing is done to scale. You should measure.
T1= 3700 lb T2= 2600 lb
Trigonometric Solution: Use the triangle rule. Note that the triangle in represents half of the parallelogram shown in. Using the law of Sines.
with a calculator, compute and store the valu of the last quotient. Multiply this value succesively by the sin 45 and sin 30. obtaining
T1=3660 lb T2= 2590lb
b. Value of a for Minimum T2.
To determine the value of a for which the tension in rope 2 is minimum, use the triangle rule again. In line 1-1' is the known direction of T1. Several possible directions of T2 are shown by the lines 2-2'. The minimum value of T2 occurs when T1 and T2 are perpendicular. Thus, the minimum value of T2 is
Statics of Particles
Introduction•The objective for the current chapter is to investigate the effects of forces on particles:
-replacing multiple forces acting on a particle with a single equivalent or resultant force,
-relations between forces acting on a particle that is in a state of equilibrium.
•The focus on particles does not imply a restriction to miniscule bodies. Rather, the study is restricted to analyses in which the size and shape of the bodies is not significant so that all forces may be assumed to be applied at a single point.
Resultant of Two Forces
•force: action of one body on another; characterized by its point of application, magnitude, line of action, and sense.
•Experimental evidence shows that the combined effect of two forces may be represented by a single resultant force.
•The resultant is equivalent to the diagonal of a parallelogram which contains the two forces in adjacent legs.
•Force is a vector quantity.
Vectors
•Vector: parameters possessing magnitude and direction which add according to the parallelogram law. Examples: displacements, velocities, accelerations.•Scalar: parameters possessing magnitude but not direction. Examples: mass, volume, temperature
•Vector classifications:
-Fixed or bound vectors have well defined points of application that cannot be changed without affecting an analysis.
-Free vectors may be freely moved in space without changing their effect on an analysis.
-Sliding vectors may be applied anywhere along their line of action without affecting an analysis.
-Equal vector shave the same magnitude and direction.
-Negative vector of a given vector has the same magnitude and the opposite direction.
Parallelogram law of Forces
If two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is presented in magnitude and direction by the diagonal passing through the point.
A barge is pulled by two tugboats. If the resultant of the forces exerted by the tugboats is 5000 lbf directed along the axis of the barge, determine
a)the tension in each of the ropes for a= 45 degree,
b)the value of a for which the tension in rope 2 is a minimum.
Solution:
a. Tension for a=45 degree.
Graphical Solution: use the parallelogram law. The resultant the diagonal of the parallelogram is equal to 5000 lb and is directed to the right. Draw the sides parallel to the ropes. If the drawing is done to scale. You should measure.
T1= 3700 lb T2= 2600 lb
Trigonometric Solution: Use the triangle rule. Note that the triangle in represents half of the parallelogram shown in. Using the law of Sines.
with a calculator, compute and store the valu of the last quotient. Multiply this value succesively by the sin 45 and sin 30. obtaining
T1=3660 lb T2= 2590lb
b. Value of a for Minimum T2.
T2= 5000 sin 30 = 2500 lb
Triangle rule applied for minimum T2.
Corresponding values of T1 and a are
T1= 5000 lb cos 300
= 4330 lb
a= 90 - 30
a= 60 degree
• May resolve a force vector into perpendicular components so that the resulting parallelogram is a rectangle. are referred to as rectangular vector components and
F=Fx Fy
F=Fx Fy
• Vector components may be expressed as products of the unit vectors with the scalar magnitudes of the vector components.
F= Fx i X Fy j
Fx and Fy are referred to as the scalar components of F
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