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Work and Energy. Definitions. Work – the product of force and the component of displacement in the direction of the force. - work is a scalar quantity - Without motion there is no work. W = F ∙ d The unit of work is the newton meter, which is called a joule (J)
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Definitions • Work – the product of force and the component of displacement in the direction of the force. - work is a scalar quantity - Without motion there is no work. W = F ∙ d The unit of work is the newton meter, which is called a joule (J) (in honor of English Scientist James Prescott Joule)
Example • How much work is done on an object if a force of 30 Newtons [south] displaces the object 200 meters [south]? • Solution: W = F∙d = (30 N[S])(200 m [S]) = 6000 J
Work (cont’d) • Suppose force and displacement are not in the same direction. • Work is defined to be the product of the force in the direction of the displacement and the displacement F W = (Fcosθ)∙d d θ object
example • As Alex pulls his red wagon down the sidewalk, the handle of the wagon makes an angle of 60 degrees with the pavement. If Alex exerts a force of 100 Newtons along the direction of the handle, how much work is done when the displacement of the wagon is 20 meters along the ground?
Definition • Power – the rate at which work is done Power is also a scalar quantity, and its unit is Joules per second (J/s), also known as a watt(W).
Example • If 300 J of work is performed on an object in 1.0 minute, what is the power expended on the object? P = W / t P = 300J / 60 sec P = 5 W
Example 2 • A 200 N force is applied to an object that moves in the direction of the force. If the object travels with a constant velocity of 10m/s, calculate the power expended on the object. P = 200N (10 m/s) P = 2000 W
Hw • Read pg. 80-83 Do # 1-25
Energy Mechanical Energy
Kinetic Energy • Work done on an object changes its Kinetic Energy. • Therefore, W = KEf – KEi = ΔKE. • The formula for Kinetic Energy is KE = ½ mv2
Example • A 10 kg object subjected to a 20. N force moves across a horizontal, frictionless surface in the direction of the force. Before the force was applied, the speed of the object was 2.0m/s. When the force is removed the object is traveling at 6.0 m/s. Calculate the following quantities: (a) KEi , (b) KEf , (c)ΔKE, (d) W, and (e) d.
Solution (a) KEi = ½ mvi2 = ½ (10. kg)(2.0 m/s)2 = 20. J (b) KEf = ½ mvf2 = ½ (10. kg)(6.0 m/s)2 = 180 J
Solution (cont’d) (c) ΔKE = KEf – KEi = 180 J – 20. J = 160 J (e) W = F∙d d = W/F = 160 J / 20. N = 8.0 m (d) W = ΔKE = 160 J
Gravitational Potential Energy • An object decreases its gravitational Potential Energy (PE) as it moves closer to the Earth • To calculate the change in PE of an object we measure the work done on the object. The force needed to overcome gravity is Fg = mg. Therefore, since W = Fg ∙d , PE is defined as ΔPE = mg Δ h where Δh represents change in vertical displacement above the earth.
Example • A 2.00 kg mass is lifted to a height of 10.0 m above the surface of the Earth. Calculate the change in the PE of the object. • Solution ΔPE = mg Δ h = (2.00 kg) (9.8 m/s2)(10.0m) = 196 J
NOTE • For a change in gravitational energy to occur, there must be a change in the vertical displacement of an object; if it is moved only horizontally, the ΔPE = 0. • If an object is moved up an inclined plane, its potential energy change is measured by calculating only its verticaldisplacement; the horizontal part does not change its PE
Home work • Review book : read pg 84 Do # 26-38 (PE) • Read pg 92-93 Do #57 -66 (KE)
Conservation of Mechanical Energy • In a system, the sum of PE and KE (the total mechanical energy) is constant (i.e. conserved); a change in one is accompanied by an opposite change in the other. ΔPE = -ΔKE PEi + KEi = PEf +KEf
example • A 0.50 kg ball is projected vertically and rises to a height of 2.0 meters above the ground. Calculate: (a) the increase in the ball’s PE (b) the decrease in the ball’s KE (c) the initial KE (d) the initial speed of the ball
solution • The increase in the ball’s PE ΔPE = mg Δ h = (0.50 kg)(9.8 m/s2)(2.0 m) = 9.8 J (b) The decrease in the ball’s KE ΔKE = -ΔPE = -9.8 J
Solution cont’d (c) The initial KE • Recall that at its highest point, the speed of the ball is zero; therefore its KE is zero. So the initial KE represents the change in KE of the object . ΔKE = KEf – KEi -9.8J = 0 - KEi KEi = 9.8 J
Sol’n cont’d • The initial speed of the ball KEi = ½ mvi2 Solving for vi = sqrt (2KEi /m) = sqrt [ 2(9.8J)/(0.50kg) ] = 6.3 m/s
Pendulum Observe that the falling motion of the bob is accompanied by an increase in speed. As the bob loses height and PE, it gains speed and KE; yet the total of the two forms of mechanical energy is conserved
example • A pendulum whose bob weighs 12 N is lifted a vertical height of 0.40 m from its equilibrium position. Calculate: (a) change in PE between max height and equilibrium height (b) Gain in KE and (c) the velocity at the equilibrium point.
solution • Take PE at lowest point to be zero ΔPE = mg Δ h = FgΔ h = (12 N)(-0.40m) = -4.8 J (b) ΔKE = -ΔPE = -(-4.8 J) = 4.8 J
Sol’n cont’d ( c ) First must calculate mass of bob Fg = mg m = Fg / g = 12N / 9.8m/s2 = 1.2 kg Then calculate velocity KE= ½ mv2 v = sqrt (2KE/m) = sqrt [ 2(4.8J)/(1.2kg) ] = 2.8 m/s
Hooke’s Law • The English scientist Robert Hooke was able to show that the magnitude of a force (F) is directly proportional to the elongation (stretch) of compression of a spring (x) within certain limits. Fs = kx k – spring constant – unit is the newton per meter (N/m) Note: the greater the constant, the stiffer the spring
Elastic Potential Energy Spring is attached to a wall. If a force is applied and stretch the string to the right, work has been done. This work has been converted into the spring’s potential energy – Elastic Potential Energy PEs = ½ kx2
Hooke’s law and Potential Energy Area under the graph equals the work done. Fs = kx W
Example • A spring whose constant is 2.0 N/m is stretched 0.40 m from its equilibrium position. What is the increase in the elastic potential energy of the spring? Solution PEs = ½ kx2 = ½ (2.0 N/m ) (0.40 m)2 = 0.16 J
Elastic and Inelastic Collisions • In an elastic collision BOTH kinetic energy and momentum are conserved p1i + p2i = p1f + p2f KE1i+ KE2i = KE1f +KE2f p1i p2i p1f p2f 1 2 1 2 1 2 KE1i KE2i KE2f KE1f
Inelastic collisions • In inelastic collisions, the kinetic energy that is “lost” is converted into internal energy (Q) of the objects by frictional forces. • These systems are called nonideal mechanical systems. • The energy is constant ET = PE + KE + Q A change in the internal energy of an object is usually accompanied by a change in temperature
Simple Machines • A simple machine is a device that allows work to be done and offers and advantage to the user. Ex: pulleys, levers, inclined planes, wheels and axles and screwdrivers Win = Wout Fin ∙ din = Fout ∙ dout Fout / Fin = din / dout Mechanical Advantage (MA): Fout / Fin Ideal MA (IMA) assumes no friction (use din / dout ) Actual MA (AMA) is always less than IMA (use Fout / Fin ) The efficiency of machine is AMA/IMA ratio and this value is always less than 100%
example • A 100 N object is moved 2 m up an inclined plane whose end is lifted 0.5 m from the floor. If a force of 50 N is needed to accomplish this task, calculate the (a) IMA (b) AMA And (c) efficiency of the inclined plane
solution Input force (Fin ) = 50 N (force needed to move object) Output force (Fout ) = 100N (force that has been lifted) Input distance (din ) = 2m (distance moved along plane) Output distance (dout ) = 0.5m (distance object is raised) • IMA = din / dout = 2m / 0.5m = 4 • AMA = Fout / Fin = 100N / 50 N = 2 • efficiency = AMA/IMA = 2/4 = 0.5 (50%)
Internal Energy (Q) • Internal Energy of a system is the total kinetic and potential energies of the atoms and molecules that make up the system. • Recall: a change in the internal energy of an object is usually accompanied by a change in its temperature.
example • Force is used to move an object along a horizontal table at constant speed. • Has work been done? • Is there a change in Kinetic Energy? Why or why not? • Is there a change in Potential Energy? Why or why not? • How was work used? Yes W = F ∙ d No – speed is constant No – table is horizontal - so no change in height Used to overcome friction between object and table so internal energy of the object-table system has been increased by work done.
The Laws of Thermodynamics The study of the relationships among heat, work, and energy in the universe
The first law of Thermodynamics • Energy can neither be created nor destroyed. It can only change forms. • The change in the internal energy of a system (ΔU) is equal to the heat (Q) that the system absorbs (or releases) minus the work (W) it does (or has done on it) ΔU = Q – W
2nd Law of Thermodynamics • Result if the work of the French physicist Nicolas Carnot with heat engines. • Law states heat cannot flow from colder object to a warmer one without work being done on the system. • Ex. Refrigerators must be run by motors in order to withdraw heat from objects placed in them • No heat engine can be 100% efficient. Some of the heat absorbed by the engine must be lost in the random motion of its molecules. • Entropy is the measure of this disorder
3rd law of Thermodynamics • As temperature approaches absolute zero, (0 K) the entropy of a system approaches a constant minimum • The efficiency of a heat engine depends on its operating temperatures; engine would reach 100 % efficiency only at 0 K. • Engine cannot be completely efficient, therefore 0K cannot be reached.