PHY 101 DR M. A. ELERUJA KINETIC ENERGY AND WORK POTENTIAL ENERGY AND CONSERVATION OF ENERGY CENTRE OF MASS AND LINEAR MOMENTUM Work is done by a force acting on an object when the point of application of that force moves through some distance and the force has a component along the line of motion. kinetic energy is energy, which an object possesses because of its motion. In general, we can think of energy as the capacity that an object has for performing work. work energy concepts are based on Newton s laws and therefore allow us to make predictions that are always in agreement with these laws. Work and Kinetic Energy 7.1 Work Done by a Constant Force 7.2 The Scalar Product of Two Vectors 7.3 Work Done by a Varying Force 7.4 Kinetic Energy and the Work Kinetic Energy Theorem 7.5 Power This method of describing motion is especially useful when the force acting on a particle varies with the position of the particle, i.ethe case, in which the acceleration is not constant, and kinematic equations cannot be applied. Examine the situation in Figure 1.1, where an object undergoes a displacement d along a straight line while acted on by a constant force F that makes an angle with d.
The work W done on an object by an agent exerting a constant force on the object is the product of the component of the force in the direction of the displacement and the magnitude of the displacement: W =Fdcosθ.(1.1) A force does no work on an object if the object does not move. This can be seen by noting that if d =0, Equation 1.1 gives W = 0 The sign of the work also depends on the direction of F relative to d. The factor cosθin the definition of W (Eq. 7.1) automatically takes care of the sign. It is important to note that work is an energy transfer; if energy is transferred to the system (object), W is positive; if energy is transferred from the system, W is negative. Also note from Equation 7.1 that the work done by a force on a moving object is zero when the force applied is perpendicular to the object s displacement. That is, if 90, then W = 0 because cos90 = 0. For example, in Figure 1.2, the work done by the normal force on the object and the work done by the force of gravity on the object are both zero because both forces are perpendicular to the displacement and have zero components in the direction of d. Work is a scalar quantity, and its units are force multiplied by length. Therefore, the SI unit of work is the newtonmeter(nm) = joule (J).
THE SCALAR PRODUCT OF TWO VECTORS Work can be expressed as the scalar product, F d, also known as dot product W = F d= Fdcosθ (1.2) In general, the scalar product of any two vectors A and B is a scalar quantity equal to the product of the magnitudes of the two vectors and the cosine of the angle between them: AB = AB cosθ The scalar product is commutative. That is, Finally, the scalar product obeys the distributive law of multiplication, so that The dot product is simple to evaluate from Equation 1.3 when A is either perpendicular or parallel to B. If A is perpendicular to B ( θ= 90 ), then A B = 0. (The equality A B = 0 also holds in the more trivial case when either A or B is zero.) If vector A is parallel to vector B and the two point in the same direction( θ= 0), then A B = AB. If vector A is parallel to vector B but the two point in opposite directions ( θ= 180 ), then A B = AB. The scalar product is negative when 90 < θ< 180. In Figure 1.3, B cosθ is the projection of B onto A. Therefore, Equation 1.3 says that AB is the product of the magnitude of A and the projection of B onto A. The unit vectors i, j, and k, which lie in the positive x, y, and z directions, respectively, are of a right-handed coordinate system. Therefore, it follows from the definition of A B that the scalar products of these unit vectors are i i= j j= k k= 1..(1.4) i j= j k= k i= 0 (1.5)
In the special case where A = B, A.A is given by Work done by a constant force A particle moving in the x-y plane undergoes a displacement d = (3i + 4j + 5k) m as a constant force, F = (2i -4j + 5k) N. Calculate the (i) magnitude of the displacement and the force and (ii) the work done by the force. (i) (ii) W = [(3)(2) + (4)(4) + (5)(5)] = [6+16+25] = 47 J If the displacements are allowed to approach zero, then the number of terms in the sum increases without limit but the value of the sum approaches a definite value equal to the area bounded by the F x curve and the x axis: This definite integral is numerically equal to the area under the F x -versus-x curve between xi and x f. Therefore, we can express the work done by F x as the particle moves from x i to x f as (1.6) Work done by a varying Force a b If more than one force acts on a particle, the total work done is just the work done by the resultant force. If we express the resultant force in the x direc on as F x, then the total work, or net work, done as the particle moves from x i to x f is
Work Done by a Spring. A common physical system for which the force varies with position is that in which a block on a horizontal, frictionless surface is connected to a spring. If the spring is either stretched or compressed a small distance from its unstretched(equilibrium) configuration, it exerts on the block a force of magnitude Suppose the block has been pushed to the left a distance x max from equilibrium and is then released. Let us calculate the work W s done by the spring force as the block moves from x i = x max to x f =0. Applying Equation 1.6 and assuming the block may be treated as a particle, we obtain KINETIC ENERGY AND THEWORK KINETIC ENERGY THEOREM It is difficult to use Newton s laws to solve motion problems involving complex forces. An easy alternative is to relate the speed of a moving particle to its displacement under the influence of some net force. According to Newton s second law of motion, since the particle is acted upon by a constant force, it experiences a constant acceleration. If the particle is displaced a distance d, the net work done by the total force F is Recall that the following relationships are valid when a particle undergoes constant acceleration:
where v i is the speed at t = 0 and v f is the speed at time t. Substituting these expressions into Equation 1.11 gives The quantity represents the energy associated with the motion of the particle. This quantity is so important that it has been given a special name kinetic energy. The net work done on a particle by a constant net force F ac ng on it equals the change in kinetic energy of the particle. From this theorem, Note that: (i) the speed of a particle increases if the net work done on it is positive because the final kinetic energy is greater than the initial kinetic energy. (ii) The particle s speed decreases if the net work done is negative because the final kinetic energy is less than the initial kinetic energy. The work kinetic energy theorem was derived under the assumption of a constant net force, but it also is valid when the force varies. Conclusion: The net work done on a particle by the net force acting on it is equal to the change in the kinetic energy of the particle. This is true whether or not the net force is constant. In general, the kinetic energy K of a particle of mass m moving with a speed v is defined as Kinetic energy is a scalar quantity and has the same units as work. It is often convenient to write Equation 1.12 in the form That is K i + W = K f Equation 1.14 is known as the Work-kinetic energy theorem POWER The time rate of doing work is called power. If an external force is applied to an object (which we assume acts as a particle), and if the work done by this force in the time interval Δtis W, then the average power expended during this interval is defined as A more general definition of power is the time rate of energy transfer.
In a manner similar to velocity and acceleration, the instantaneous power is defined, Ƥ as the limiting value of the average power as Δt approaches zero: The instantaneous power can be written as The SI unit of power is joules per second (J/s), also called the watt (W) Potential Energy Another form of energy, potential energy, is the energy associated with the arrangement of a system of objects that exert forces on each other. Potential energy can be thought of as stored energy that can either do work or be converted to kinetic energy. The potential energy concept can be used only when dealing with a special class of forces called conservative forces Potential Energy and Conservation of Energy 2.1 Potential Energy 2.2 Conservative and Nonconservative Forces 2.3 Conservative Forces and Potential Energy 2.4 Conservation of Mechanical Energy 2.5 Work Done by Nonconservative Forces 2.6 Relationship Between Conservative Forces and Potential Energy 2.7 Conservation of Energy in General If the arrangement of the system changes, then the potential energy of the system changes. Gravitational Potential Energy The product of the magnitude of the gravitational force mg acting on an object and the height y of the object is so important in physics that we give it a name: the gravitational potential energy. The symbol for gravitational potential energy is U g, and so the defining equation for gravitational potential energy is
Gravitational potential energy is the potential energy of the object Earth system. For a system of two objects, such as the object Earth, in which the Earth (may be assumed to stationary) is far more massive than the object, then kinetic energy of the system is represented by the kinetic energy of the object falling toward the Earth. Finally, the work done by the gravitational force on an object as the object falls to the Earth is the same as the work done were the object to start at the same point and slide down an incline to the Earth. Horizontal motion does not affect the value of W g. The unit of gravitational potential energy is the same as that of work the joule. Potential energy, like work and kinetic energy, is a scalar quantity. The work done on any object by the gravitational force is equal to the negative of the change in the system s gravitational potential energy. Note that: It is only the difference in the gravitational potential energy at the initial and final locations that matters. Elastic Potential Energy For a system consisting of a block and a spring, the work done by spring on the attached block is The elastic potential energy of the system can be thought of as the energy stored in the deformed spring (one that is either compressed or stretched from its equilibrium position).
CONSERVATIVE AND CONCONSERVATIVE FORCES Conservative forces have two important properties: 1. A force is conservative if the work it does on a particle moving between any two points is independent of the path taken by the particle. 2. The work done by a conservative force on a particle moving through any closed path is zero. (A closed path is one in which the beginning and end points are identical.) Examples of Conservative force are (i) The gravitational force and (ii) the force that a spring exerts on any object attached to the spring. Nonconservative Forces A force is nonconservativeif it causes a change in mechanical energy E, which we define as the sum of kinetic and potential energies. For example, if a book is sent sliding on a horizontal surface that is not frictionless, the force of kinetic friction reduces the book s kinetic energy. For the case of the object spring system, the work W s done by the spring force is given by (Eq. 2.3). The spring force is conservative because W s depends only on the initial and final x coordinates of the object and is zero for any closed path. A potential energy with any conservative force and can do this only for conservative forces. In the previous section, the potential energy associated with the gravitational force was defined as in general, the work W c done on an object by a conservative force is equal to the initial value of the potential energy associated with the object minus the final value: As the book slows down, its kinetic energy decreases. As a result of the frictional force, the temperatures of the book and surface increase. The type of energy associated with temperature is internal energy. It should be noted that this internal energy cannot be transferred back to the kinetic energy of the book, that is, the energy transformation is not reversible. Because the force of kinetic friction changes the mechanical energy of a system, it is a nonconservative force.
From the work kinetic energy theorem, we see that the work done by a conservative force on an object causes a change in the kinetic energy of the object. The change in kinetic energy depends only on the initial and final positions of the object, and not on the path connecting these points. Let us compare this to the sliding book example, in which the nonconservativeforce of friction is acting between the book and the surface. The change in kinetic energy of the book due to friction is, ΔK friction = -f k d where d is the length of the path over which the friction force acts. Thus, we see that for a conconservativeforce, the change in kinetic energy depends on the path followed between the initial and final points. If a potential energy is involved, then the change in the total mechanical energy depends on the path followed. Consider that the book slides from A to B over the straight-line path of length d in Figure 2.3. The change in kinetic energy is -f k d. Now, suppose the book slides over the semicircular path from A to B. In this case, the path is longer and, as a result, the change in kinetic energy is greater in magnitude than that in the straight-line case. For this particular path, the change in kinetic energy is, -f k πd/2,since d is the diameter of the semicircle. Thus, we see that for a nonconservativeforce, the change in kinetic energy depends on the path followed between the initial and final points. If a potential energy is involved, then the change in the total mechanical energy depends on the path followed. CONSERVATIVE FORCES AND POTENTIAL ENERGY The work done on a particle by a conservative force does not depend on the path taken by the particle. The work depends only on the particle s initial and final coordinates.
The work depends only on the particle s initial and final coordinates. Consequently, we can define potential energy function U such that the work done by a conservative force equals the decrease in the potential energy of the system. The work done by a conservative force F as a particle moves along the x axis is The term potential energy implies that the object has the potential, or capability, of either gaining kinetic energy or doing work when it is released from some point under the influence of a conservative force exerted on the object by some other member of the system. It is often convenient to establish some particular location x i as a reference point and measure all potential energy differences with respect to it. where F x is the component of F in the direction of the displacement. That is, the work done by a conservative force equals the negative of the change in the potential energy associated with that force, where the change in the potential energy is defined as ΔU = U f U i and Equation 2.4 can be expressed as ( 2.5 ) We can then define the potential energy function as The value of U i is often taken to be zero at the reference point. It really does not matter what value we assign to U i, because any nonzero value merely shifts U f (x) by a constant amount, and only the change in potential energy is physically meaningful.
CONSERVATION OFMECHANICAL ENERGY An object held at some height h above the floor has no kinetic energy. However, as the gravitational potential energy of the object Earth system is equal to mgh. If the object is dropped, it falls to the floor; as it falls, its speed and thus its kinetic energy increase, while the potential energy of the system decreases. If factors such as air resistance are ignored, whatever potential energy the system loses as the object moves downward appears as kinetic energy of the object. In other words, the sum of the kinetic and potential energies the total mechanical energy E remains constant. This is an example of the principle of conservation of mechanical energy. Also: Principle of conservation of energy as E i = E f canbe expressed as If more than one conservative force acts on an object within a system, a potential energy function is associated with each force. In such a case, we can apply the principle of conservation of mechanical energy for the system as where the number of terms in the sums equals the number of conservative forces Note that: The total mechanical energy of a system remains constant in any isolated system of objects that interact only through conservative forces. Because the total mechanical energy E of a system is defined as the sum of the kinetic and potential energies, we can write Linear Momentum and Collisions 9.1 Linear Momentum and Its Conservation 9.2 Impulse and Momentum 9.3 Collisions 9.4 Elastic and Inelastic Collisions in One Dimension 9.5 Two-Dimensional Collisions 9.6 The Center of Mass 9.7 Motion of a System of Particles
LINEAR MOMENTUM AND ITS CONSERVATION The concept of momentum,is useful for describing objects in motion and as an alternate and more general means of applying Newton s laws. The linear momentum of a particle of mass m moving with a velocity v is defined to be the product of the mass and velocity: P mv (9.1) Linear momentum is a vector quantity because it equals the product of a scalar quantity m and a vector quantity v. Its direction is along v, it has dimensions ML/T, and its SI unit is kg m/s. Using Newton s second law of motion, we can relate the linear momentum of a particle to the resultant force acting on the particle: The time rate of change of the linear momentum of a particle is equal to the net force acting on the particle: If a particle is moving in an arbitrary direction, p must have three components, and Equation (3.1) is equivalent to the component equations P x = mv x, P y = mv y,p z = mv z (3.2) When the net force acting on a particle is zero, the time derivative of the momentum of the particle is zero, and therefore its linear momentum1 is constant. Of course, if the particle is isolated, then by necessity and p remains unchanged.
This means that p is conserved. Just as the law of conservation of energy is useful in solving complex motion problems, the law of conservation of momentum can greatly simplify the analysis of other types of complicated motion. The impulse imparted to a particle by a force F is equal to the change in the momentum of the particle: This is known as the impulse momentum theorem. Impulsive forces are often very strong compared with other forces on the system and usually act for a very short time, as in the case of collisions. Conservation of Momentum for a Two-Particle System The law of conservation of linear momentum indicates that the total momentum of an isolated system is conserved. If two particles form an isolated system, their total momentum is conserved regardless of the nature of the force between them. Therefore, the total momentum of the system at all times equals its initial total momentum, or.