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Transcript of Physics
Fs = Us x Fn Fs is the force of static friction on the object, Us is the coefficient of friction and Fn is the normal force of the surface on the object
Static frictional force acts in the plane of the surface, and only exists when there is an eternal, opposite force on the object. Equilibrium is Small Objects Small objects, in comparison to large objects, are easier to analyze because all forces on them act at essentially the same point. Large objects can be treated the same as small objects if we consider all the forces to act through a single point called the center of mass. The center of mass for a uniform, regular object is the geometric center of that object.
In order for an object to be in equilibrium, the sum of all the forces acting on that object must be zero. Therefore the net force must be 0. Introduction Equilibrium describes the forces acting in such a way that the net force on an object is zero. An object in equilibrium and at rest is said to be in static equilibrium, whereas an object moving with an uniform velocity is said to be in dynamic equilibrium Static Equilibrium in Large Bodies
Torque An object can only be in equilibrium if the sum of all the forces acting on it is zero. However, this is an incomplete condition.
Two children pull a log across the ice. Their ropes are attached one to each end of the log and they are moving in a northerly direction with the log parallel to the ropes. After stoping for arrest, one child decides to go west and pulls with a force of 75N due west. The other child disagrees, and pulls with a force of 75 N due east. Center of Gravity An object can be considered to be made up of a large number of tiny, equal particles, each of which is pulled to the Earth. Stable, Unstable, and Neutral Equilibrium Ship and car designers must take stability into account during the design stage. Ships, in bad weather, can turn over and sink. A car travelling too fast while turning can also overturn. These objects must be designed so that they have stable equilibrium under most conditions.
The stability of an object depends on the position of its centre of gravity and the torque its weight exerts. by Valavan Manan Adnan Brandon Equilibrium Static equilibrium is observed during all parts of our day. For example, when we stand up, the upward forces of the floor on our feet combine with the downward forces of gravity to keep the body in a steady state. Snow on our roof is under equilibrium, or even a ladder resting against a wall. Lets not let anymore of these disasters happen! The knowledge gained from studying static equilibrium is important to architects and engineers. They have to be able to calculate stresses and loads in the structural parts of a bridge or building. Anything can buckle and break if too much force is applied to it. Did you know? The Quebec Bridge collapse was due to poor judgement of the engineers. The natural weight of the bridge was far too heavy to carry anything other than itself. So the bridge collapsed killing seventy-five workers. Coaches often advise linemen to stay low. This brings their center of mass closer to the ground, so an opposing player, no matter how low he goes, can only contact them near their center of mass. This makes it difficult for an opposing player to move them, as they will not rotate upon contact. This technique is critical for a defensive lineman in defending his own goal in the "red" zone, the last 10 yards before the goal line. Say Hut! From this, it is clear that the net force on the log is zero. But the log will not remain stationary. It will rotate with each end moving in the opposite direction. The log is still under static equilibrium since the log’s center of mass is the pivot point, and this remains at rest.
This shows that in large objects, the forces involved may not pass through a common point as they do for small objects. We must also take the twisting or turning effect of each force into account. This twisting effect is called torque or the moment of the force. Three factors are important in determining the torque:
1) The size of the force applied
2)The point of application
3)The direction in which the force is applied
T = F x r T is the torque, F is the applied force, and r is the perpendicular distance between the point of rotation and the line of action of the applied force.
The unit for Torque is Newton meter (N*m) and t. The symbol usually used for it is T. If the twisting effect is counter clockwise, it is called positive torque. If it is clockwise, it is negative torque. When two or more torques act on an object, each will tend to rotate the object about a given axis. For static equilibrium, the sum of the counter clockwise torques about any point is equal to the sum of the clockwise torques. This is called the Principle of Torques, but is also referred to as the Principle of Moments or the Law of the Lever. Now, we have two fundamental conditions that must be satisfied if any object, large or small, is to remain in static equilibrium :
1) The net force must be zero, no matter what direction is chosen.
2) The total torque acting on a rigid object is zero, about any chosen axis of rotation. For example, the earth’s pull on a rock consists of a large number of equal, parallel forces, each acting on the particles that make up the rock.
These forces have a vertical resultant force downward which equals to the sum of the individual forces acting on the particles. This net downward force acts through a point called the center of gravity. The center of gravity of a body is the point of application of the resultant force representing the force of gravity on the object. The center of gravity for long narrow objects, like rulers, is directly above the point of support when the object is balanced. The particles on either side of the balance point each have torque, but the net torque is zero, since the object doesn’t rotate.
For objects with uniform composition, the center of gravity can be determined geometrically. For example, the center of gravity of a solid cylinder lies at the midpoint of the cylinder’s axis. Difference between centre of mass and centre of gravity? Example : If you take a uniform rod and strike it with a force near its end, the rod will rotate as it accelerates. On the other hand, if the rod is struck -at its centre it will accelerate without rotation. The centre of mass of an object may be defined as the point at which a single applied force produces acceleration, but no rotation. In the case of the rod, the centre of mass and gravity are at the same position. This is true for a majority of objects. Unstable equilibrium When the center of gravity of a body lies above the point of suspension or support, the body is said to be in unstable equilibrium. For example, a pencil standing on its point is said to be in unstable equilibrium. It is in unstable equilibrium because if the pencil is slightly disturbed, it will not come back to its original position. When the pencil is disturbed, its centre of gravity is lowered. The line of centre of gravity lies outside the base of the pencil. Therefore, the torque due to the weight of the pencil topples it down. Stable equilibrium When the centre of gravity of a body lies below the point of support, the body is said to be in stable equilibrium. For example, a book lying on a table is in stable equilibrium. If the book is lifted from one edge, and then allowed to fall, it will come back to its original position. When the book is lifted, its centre of gravity is raised. The centre of gravity passes through the base of the book. The torque due to the weight of the book brings it back to the original position. Neutral equilibrium When the centre of gravity of a body lies at the point of suspension or support, the body is said to be in neutral equilibrium. For example, a rolling ball is said to be in neutral equilibrium. If a ball is pushed slightly, it will neither come back to its original position, nor will it roll forward, it will remain at rest. The centre of gravity of the ball is neither raised nor lowered. Therefore, the centre of gravity is at the same height as before. As discussed earlier, when the vertical line from the centre of gravity of an object lies outside its base, an object is unstable and will tip over. The critical position for beginning to tip is called the critical tipping angle. This is the position when the centre of gravity is directly above the pivot point. Hooke’s Law We calculated the forces acting on objects in equilibrium. Now, we will study the effects of these forces on the objects. This is important to do because, by doing so, we can determine whether the forces are great enough to cause an object to break, fracture, or be deformed.
When forces are applied to an object, the dimensions of the object tend to change. For example, if a force is applied to a spring, it either stretches or compresses. When the force is removed, the spring returns to its original length. If an object returns to its original orientation after the applied force is removed, that object is elastic. Robert Hooke was one of the first to study the elasticity of objects. In 1678, he came up with what is now known as Hooke’s Law. It states that the amount of deformation of an elastic object is proportional to the forces applied to deform it. Young’s Modulus – Stress and Strain The amount of extension of any elastic object depends on a number of other factors as well as the applied force.
1)The extension is directly proportional to the length of the elastic object (∆L L)
2)The extension is inversely proportional to the cross-sectional area (∆L 1/A)
3)The extension is directly proportional to the applied force (Hooke’s Law) (∆L F)
4)The extension is directly proportional to the constant for the material (∆L K)
Combining these proportionalities, an equation can be made.
Extension (K *F* L) / A
To change this into an equation, a constant is required. The factor in this relationship that does not vary with geometry or force is the material. The constant of proportionality for the material is called the elastic modulus or Young’s Modulus, and is given the symbol E.
∆L= (1/E) (F*L/A) or E = (F/A) / (∆L/L) mn This equation is much more useful than Hooke’s Law because it incorporates all of the factors that affect the extension of an object.
In this equation, the numerator and denominator are given special names. Force per unit area is known as stress and the ratio of change in length to the original length is called strain.
Stress = force/area = F/A
Strain = change in length/original length = L/L
Therefore, E = (F/A) / ( L/L) = stress/strain. From this, it is clear that E is a constant, and strain is directly proportional to stress. There are three possible types of stress on an object – tension, compression, and shear. A stretched spring is said to be under tension or tensile stress. With tensile stress, forces act outward to cause an increase in length. On the other hand if the forces act inward on the ends of the object, causing its length to decrease we say that there is compression or compressive stress. The Young’s Modulus constant for a given material is usually the same for both tension and compression. When there is tension or compression, the forces usually act along the length of the object. For an object under shear stress, the forces act across its opposite faces. For example, if the top cover of a textbook resting on a desk is pushed with a force parallel to the surface of the desk, the shape of the book changes, although the dimensions do not. There is angular deformation.
When Young’s equation is used for shear stress, the constant of proportionality is called G, the shear modulus. Its value is usually smaller than the value of the elastic modulus, and is fiven by the following relationship:
G = (F/A) / ((L/L))
When an object is under too much stress, it may break or fracture. The maximum tensile force per unit area that an object can withstand is called its tensile strength. In conclusion, the Young’s Modulus equation is used primarily for solids. When applied to fluids, the equation is altered. If the object placed under pressure is a fluid, its volume is compressed. Pressure in a fluid is defined as force per unit area. This is the equivalent of stress. With solids, the extension is proportional to L; the volume relationship is similar. That is, extension is proportional to V (the original volume) for a specific change in pressure. The Young’s Modulus for fluids is called the bulk modulus and is written as:
B = -(F/A) / (V/V) = -P / (V/V) Real Life Application Stress and Strain in Building Construction Static Equilibrium in the Human Body The application of the principles of static equilibrium to the design and construction of buildings is very important part of the education of architects and engineers.
Historically, the first architectural innovation was the post-and-beam. This type of construction is found in ancient buildings of Egypt, Greece, China, England, and other parts of the world. It was limited to small spans, because the only materials available were stone, such as marble, and timber. When a load is place on a beam, the beam bends. The top part of the beam is in compression and the bottom part in tension. Since stone is weak in tension, the spans of ancient beams were limited and the columns had to be closely spaced. The net major development in construction was the arch. Because of the arrangement of the stones in an arch, the stress is primarily compressive. The stones push against each other to support large loads. The load is transferred along the arch to the vertical supports. Because these supports transfer horizontal as well as vertical forces, they must be very large, and /or the supporting walls have to be buttressed. The net major development in construction was the arch. Because of the arrangement of the stones in an arch, the stress is primarily compressive. The stones push against each other to support large loads. The load is transferred along the arch to the vertical supports. Because these supports transfer horizontal as well as vertical forces, they must be very large, and /or the supporting walls have to be buttressed. When iron and steel came into use, longer spans were possible, since these materials had much greater tensile strengths than earlier materials. But still the length of span was limited, because the longer the span, the deeper the beam had to be. This problem was solved by inventing the truss, which distributes a given amount of material in a very efficient manner. A truss is a triangle composed o three bars called members, which are joined together at their ends. A simple truss is used to support the roofs of houses and other narrow structures. The design of truss members is governed by the position and size of the loads on the truss. In all types of designs, the truss spreads the load over all its members, some being placed under compression and some under tension. If the truss is made up of steel, there are likely to be few problems, because steel has high tensile and compressive strengths. However, individual members must be designed so as to prevent buckling under compression. Wooden trusses of comparable size are not as strong as steel. Therefore, the strength of wooden trusses can be increasing tensioning the trusses. By tensioning the trusses, the wood is compressed, thus increasing strength and the tensile capacity. This concept of tensioning is most commonly used today with concrete. Concrete is strong in compression but weak in tension. As a result, it is very suitable for vertical, supporting walls and foundations, where the structure is under compression. However, its use for beams is limited, since its resistance to tensions is very low. The three parts of the human body responsible for physical movement are the muscles, the tendons, and the bones. Muscle cells are arranged in long fibres that generate contractive forces as they decrease in length and increase in width. Only the skeletal muscles are under conscious control. These muscles produce the motion of the body. The tendons transfer tensile forces of these muscles to the bones. Bones support the body and maintain its shape. They must be stiff, and strong in both compression and tension. They act as levers, converting the pulls of the muscles into both pushed and pulls. Bones have properties similar to reinforced concrete. They are made up of two components : calcium crystals and collagen fibres. The calcium crystals act like concrete, carrying high compressive loads, though they are weak in tension. The tensile load is carried by collagen fibres, which act like the steel in reinforced concrete. To compare, the tensile strength of the human bone is 1.3 x 10^8 N/m^2 while the tensile strength of iron is 1.7 x 10^8 N/m^2. Bones, however, are not solid throughout. The central region of the bone is hollow or full of bone marrow. Because it is hollow, a bone is much stronger in resisting torsion and bending than a solid bone of the same mass would be.
Accidents often apply large forces that exceed the tensile, compressive or shear strengths of a bone, and fracture occurs. The forces causing the fracture can be tensile, compressive, shearing, or a combination of these. The Elbow:
Since muscles only exert tension, two muscles are required to raise and lower the forearm. When the triceps muscle, which runs from the upper end of the forearm to the shoulder, contracts, the arm straighten out. When the biceps muscle, which is connected to the shoulder and the radius bone of the forearm, the arm bends at the elbow.
The forearm acts like a lever. When the forearm is perpendicular to the upper arm, the torque of the triceps muscle opposes the torque of the biceps muscle. The Human Foot:
When a person stands on his or her toes, the foot acts like a lever. The Achilles tendon pulls upwards, and the tibia exerts a downward force on the joint at the ankle. At the toes, the reaction force of the ground pushes upward with a force equal to the body weight carried by that leg. Equilibrium
by Valavan, Manan, Adnan and Brandon Physics 2012
Mr. Bhatia Sample Question A horizontal pull is being exerted on a 10kg box with a string. If the coefficient of static friction is 0.4, what is the maximum force in the string just before the box moves?
Fs = Us x m x g
= 39 N
Sample Question What is the tipping angle for a 2kg brick, 6cm by 12cm by 20cm when it is place on end?
Since the brick is a uniform object, its centre of gravity will be located at its geometric centre. Also, at the critical tipping angle, the centre of gravity will be directly above the edge. Given the dimensions in the diagram, it follows that tanx = 6cm/10cm, x = 31 degrees.