Work and Energy. Work joule Hooke s Law Spring Equilibrium kinetic energy work energy principle translational kinetic energy

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1 Work and Energy Vocabulary Work joule Hooke s Law Spring Equilibrium kinetic energy work energy principle translational kinetic energy 7-1 Work done by a constant force We will now also discuss the alternative analysis of motion of an object in terms of energy and momentum The significance of these quantities is that they are conserved- they remain constant by conserving these quantities we are better able to attach the problems The conservation laws of energy and momentum are especially valuable in dealing with systems of many objects in which a detailed consideration of the forces involved would be difficult or impossible These laws work for the subatomic world and on up (Newtons laws so not always work in the subatomic world) Energy and work are closely related to one another and are scalar quantities- no direction associated with them Energy is important because it is conserved and it is useful in studying motion work is an action of force when it acts on an object over some distance. Work done on an object by a constant force (constant in both magnitude and direction) is defined to be the product of the magnitude of the displacement times the component of the force parallel to the displacement Equation Box 7-1 F is parallel to the displacement d Equation Box 7-2 F is the magnitude of the constant force, d is the magnitude of the displacement of the object, and ø is the angle between the directions of the force and the displacement Work is a scalar quantity- it only has magnitude 1 RoessBoss

2 Units for work: the joule 1J= 1Nm a Force can be exerted on an object and yet do no work. Carrying groceries is no work (displacement is zero) When dealing with work, as with force, it is necessary to specify whether you are talking about work done by a specific object or done on a specific object It is also important to specify whether the work done is due to one particular force (and which on), or the total (net) work done by the net force on the object 7-2 Scalar Products of two vectors Although work is scalar is involves the product of two quantities, force and displacement, each which are vectors Vectors have direction and magnitude- they can not be multiplied the same way scalars are Instead we must define what the operation of vector multiplication means Three ways in physics o multiplication of a vector by a scalar (discussed in Kinematics 2D) o multiplication of one vector by a second vector so as to produce scalar (scalar product) o multiplication of one vector by a second vector so as to produce another vector (vector product) Scalar product (dot product)- a dot is used to indicate the multiplication- scalar product Equation Box 7-3 A and B are the magnitudes of the vectors and ø is the angle (<180 ) between them when their tails touch A and B and cos ø are scalars so is the product A B Equation Box 7-4 The definition of scalar product is choose because many physically important quantities such as work can be described as the scalar product of the two vectors You can also say that it is the product of the magnitude of one vector and the component of the other vector along the direction of the first 2 RoessBoss

3 It does not matter the order in which you multiply them- scalar product is commutative It is also easy to show that it is distributive also 7-3 Work done by a Varying Force Many cases the force varies in magnitude or direction during a process This is where calculus comes into play Just as before we let the limit approach o and will take the integral The work done by a variable force in moving an object between two points is equal to the area under the F cos ø versus l curve between those two points Equation Box 7-5 This is the most general definition of work We use this concept to determine the Force on a spring When a spring is compressed or stretched an amount x from its normal length requires a force Fp that is directly proportional to x Equation Box 7-6 k is a constant, called the spring constant and is a measure of the stiffness of the particular spring The spring itself exerts a force in the opposite direction Equation Box 7-7 This force is sometimes called the restoring force because the spring exerts its force in the direction opposite the displacement (hence the minus sign), acting to return to its normal length Equation 7-7 is known as the spring equation and also as Hooke s Law. It works as long as x is not too great 7-4 Kinetic Energy and the Work Energy Theorem 3 RoessBoss

4 Energy is one of the most important concepts in science The total energy is the same after any process occurs as it was before; that is, the quantity energy is the same after any process occurs as it was before; that is, the quantity energy can be defined so that it is a conserved quantity Energy can be defined as the ability to do work It is not precise or valid for all types of energy a moving object can do work on another object it strikes a flying cannonball does wok on a brick wall it knocks down an object in motion has the ability to do work and thus can be said to have energy the energy of motion is called kinetic energy from the greek work kinetikos motion Equation Box 7-8 Then we define the quantity to be K or translational kinetic energy (1/2mv^2) Equation Box 7-9 We can rewrite this as Equation Box 7-10 The net work done on an object is equal to the change in its kinetic energy- Work Energy Principle We made use of Newton s second law and consider all the forces The work energy principle is valid only if W is the net work done on the objectthat is the work done by all forces acting on the object This principle tells us that if (positive) net work W is done on a body, kinetic energy increases by an amount W The principle also holds true for the reverse situation; if negative net work W is done on the body, the body s kinetic energy decreases by an amount W That is a net force exerted on a body opposite to the body s direction of motion reduces its speed and its kinetic energy 4 RoessBoss

5 To summarize the connection between work and kinetic energy operates both ways If the net work W done on an object is positive, then the objects kinetic energy increases If the net work W done on an object is negative, its kinetic energy decreases If the net work done on the object is zero, its kinetic energy remains constant (speed is also constant) Force parallel to the motion contributes to the work a force acting perpendicular to the velocity vector does no work Such a force changes only the direction of the velocity It does not affect the magnitude of the velocity Centripetal force does no work This force does no work on the object, because it is always perpendicular to the objects displacement Because of the direct connection between work and kinetic energy, energy is measured in the same unit as work; joules in SI units Like work kinetic energy is a scalar quantity The kinetic energy of a set of objects is the scalar sum of the kinetic energies of the individual objects Kinetic energy at very high speeds- Einstein predicted that no particle can exceed the speed of light 5 RoessBoss