Equilibrium 1x4h6b
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Equilibrium, Indeterminate Structure
Objectives 1. Introduction 2. Equilibrium 3. The Conditions for Equilibrium
Introduction • Human construction are supposed to be stable on spite of the forces that act on it.
• One focus of physics is on what allows the object to be stable in spite of any forces acting on it. • Equilibrium • Elasticity
The Leaning Tower of Pisa
•
•
The tower began to sink after construction had progressed to the second floor in 1178. This was due to a mere three-metre foundation, set in weak, unstable subsoil Prior to restoration work performed between 1990 and 2001, the tower leaned at an angle of 5.5 degrees, but the tower now leans at about 3.99 degrees.[
Sabah Foundation Building, a circular structure that is ed by high tensile and steel rods, about 30 storey's high.
Facts • 1. The linear momentum mass is constant.
of its center of
• 2. Its angular momentum about its center of mass, or about any other point, is also zero. •
Equilibrium Conditions
Conclusion If a body is in equilibrium: (1) The vector sum of all the external forces that act on it must be zero; and (2) the vector sum of all the external torques that act on it, measured about any possible point, must also be zero.
Governing Formulas
Center of Mass • Facts: • 1. The gravitational force on the body acts a single point, called the center of gravity of the body • 2. If the g is the same for all elements of a body’s center gravity (cog) is coincident with the body’s center of mass.
Review of the Topic on Concurrent Forces • 1. Forces Convention • 2. Torque • 3. Summation of Torque
Suggested Procedures • 1. Draw the equivalent free body diagram. • 2. Apply the governing formulas
Free Body Diagram • 1. Weight is always downward • 2. The direction of tension is away from the body of consideration • 3. The direction of compression is towards the body of consideration. • 4. The normal force is always exerted by the ground toward the center of gravity. • 5. Hinges provides two reactions Fx and Fy.
Free Body diagram Draw the free body diagram of the following body
60°
30°
30°
30°
Illustrative Example 1 Weights are attached to 8 meter long levers at rest. Determine the unknown weights below
??
20 N
??
20 N
20 N
??
Illustrative Example 2 A boy and his cat sit on a seesaw. The cat has a mass of 4 kg and sits 2 m from the center of rotation. If the boy has a mass of 50 kg, where should he sit so that the see-saw will balance?
Illustrative Example 3 Two work men are carrying a 6.0-m-long plank as shown in Figure. The plank has a mass of 15 kg. A cat, with a mass of 5.0 kg, jumps on the plank and hangs on, 1.0 m from the end of the plank. Assuming that the workers are walking at a constant velocity, how much force does each workman have to exert to hold the plank up?
#7/p.321 A diver of weight 580N stands at the end of a diving board of length L= 4.5m and negligible mass. The board is fixed to two pedestals separated by distance d=1.5m. Of the forces acting on the board, what are the a) magnitude, and b)direction(up or down) of the force from the left pedestal and c)magnitude, and d)direction(up or down) of the force from the right pedestal.
EXAMPLE 2: Fireman on a Ladder In Figure a, a ladder of length L = 12 m and mass m = 45 kg leans against a slick (frictionless) wall. Its upper end is at height h = 9.3 m above the pavement on which the lower end rests (the pavement is not frictionless). The ladder's center of mass is L/3 from the lower end. A firefighter of mass M = 72 kg climbs the ladder until her center of mass is L/2 from the lower end. What then are the magnitudes of the forces on the ladder from the wall and the pavement?
#12/p.322 The system is in equilibrium, with the string in the center exactly horizontal. Block A weighs 40N, block B weighs 50N, and angle Ф is 35⁰, find a) tension T1 b) T2 c) T3 and d) θ
27/p.323 The system in the figure is in equilibrium. A concrete block of mass 225 kg hangs from the end of a uniform strut of mass 45.0 kg. For angles Φ= 30.0⁰ and θ = 45.0⁰, find a)the tension T in the cable and b) the horizontal and c) the vertical components of the force on the strut from the hinge.
Seat Work • SW # 1
SW # 2
SW # 3
#30/p.324 Suppose the length L of the uniform bar is 3.00 m and its weight is 200N. Also let the block’s weight W= 300N and the angle θ = 30.0⁰. The wire can withstand a maximum tension of 500N. A) What is the maximum possible distance x before the wire breaks? B) With the block placed at this maximum x, what are the horizontal and C) vertical components of the force on the bar from the hinge at A?
Objectives 1. Introduction 2. Equilibrium 3. The Conditions for Equilibrium
Introduction • Human construction are supposed to be stable on spite of the forces that act on it.
• One focus of physics is on what allows the object to be stable in spite of any forces acting on it. • Equilibrium • Elasticity
The Leaning Tower of Pisa
•
•
The tower began to sink after construction had progressed to the second floor in 1178. This was due to a mere three-metre foundation, set in weak, unstable subsoil Prior to restoration work performed between 1990 and 2001, the tower leaned at an angle of 5.5 degrees, but the tower now leans at about 3.99 degrees.[
Sabah Foundation Building, a circular structure that is ed by high tensile and steel rods, about 30 storey's high.
Facts • 1. The linear momentum mass is constant.
of its center of
• 2. Its angular momentum about its center of mass, or about any other point, is also zero. •
Equilibrium Conditions
Conclusion If a body is in equilibrium: (1) The vector sum of all the external forces that act on it must be zero; and (2) the vector sum of all the external torques that act on it, measured about any possible point, must also be zero.
Governing Formulas
Center of Mass • Facts: • 1. The gravitational force on the body acts a single point, called the center of gravity of the body • 2. If the g is the same for all elements of a body’s center gravity (cog) is coincident with the body’s center of mass.
Review of the Topic on Concurrent Forces • 1. Forces Convention • 2. Torque • 3. Summation of Torque
Suggested Procedures • 1. Draw the equivalent free body diagram. • 2. Apply the governing formulas
Free Body Diagram • 1. Weight is always downward • 2. The direction of tension is away from the body of consideration • 3. The direction of compression is towards the body of consideration. • 4. The normal force is always exerted by the ground toward the center of gravity. • 5. Hinges provides two reactions Fx and Fy.
Free Body diagram Draw the free body diagram of the following body
60°
30°
30°
30°
Illustrative Example 1 Weights are attached to 8 meter long levers at rest. Determine the unknown weights below
??
20 N
??
20 N
20 N
??
Illustrative Example 2 A boy and his cat sit on a seesaw. The cat has a mass of 4 kg and sits 2 m from the center of rotation. If the boy has a mass of 50 kg, where should he sit so that the see-saw will balance?
Illustrative Example 3 Two work men are carrying a 6.0-m-long plank as shown in Figure. The plank has a mass of 15 kg. A cat, with a mass of 5.0 kg, jumps on the plank and hangs on, 1.0 m from the end of the plank. Assuming that the workers are walking at a constant velocity, how much force does each workman have to exert to hold the plank up?
#7/p.321 A diver of weight 580N stands at the end of a diving board of length L= 4.5m and negligible mass. The board is fixed to two pedestals separated by distance d=1.5m. Of the forces acting on the board, what are the a) magnitude, and b)direction(up or down) of the force from the left pedestal and c)magnitude, and d)direction(up or down) of the force from the right pedestal.
EXAMPLE 2: Fireman on a Ladder In Figure a, a ladder of length L = 12 m and mass m = 45 kg leans against a slick (frictionless) wall. Its upper end is at height h = 9.3 m above the pavement on which the lower end rests (the pavement is not frictionless). The ladder's center of mass is L/3 from the lower end. A firefighter of mass M = 72 kg climbs the ladder until her center of mass is L/2 from the lower end. What then are the magnitudes of the forces on the ladder from the wall and the pavement?
#12/p.322 The system is in equilibrium, with the string in the center exactly horizontal. Block A weighs 40N, block B weighs 50N, and angle Ф is 35⁰, find a) tension T1 b) T2 c) T3 and d) θ
27/p.323 The system in the figure is in equilibrium. A concrete block of mass 225 kg hangs from the end of a uniform strut of mass 45.0 kg. For angles Φ= 30.0⁰ and θ = 45.0⁰, find a)the tension T in the cable and b) the horizontal and c) the vertical components of the force on the strut from the hinge.
Seat Work • SW # 1
SW # 2
SW # 3
#30/p.324 Suppose the length L of the uniform bar is 3.00 m and its weight is 200N. Also let the block’s weight W= 300N and the angle θ = 30.0⁰. The wire can withstand a maximum tension of 500N. A) What is the maximum possible distance x before the wire breaks? B) With the block placed at this maximum x, what are the horizontal and C) vertical components of the force on the bar from the hinge at A?
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