Chatper 7 - Kinetic Energy and Work
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1 Chatper 7 - and Energy and Examples The release of atomic energy has not created a new problem. It has merely made more urgent the necessity of solving an existing one. - Albert Einstein David J. Starling Penn State Hazleton PHYS 211
2 Energy and Energy is a scalar quantity that describes the current status of one or more objects and can take many forms. Examples
3 Energy and Energy is a scalar quantity that describes the current status of one or more objects and can take many forms. Examples Energy is conserved but can transform from one type to another.
4 Energy and Kinetic energy is the energy of motion and is defined for an object to be Examples K = 1 2 mv2.
5 Energy and Kinetic energy is the energy of motion and is defined for an object to be Examples K = 1 2 mv2. Heavier and faster objects carry more energy.
6 Energy and The units of energy are (from mv2 ) kg - m2 /s2 which is given the name joule (J). Examples
7 Energy and The units of energy are (from mv2 ) kg - m2 /s2 which is given the name joule (J). James Prescott Joule, not the Crowned Jewels. Examples
8 Energy and Lecture Question 7.1 To see why professional baseball pitchers are remarkable, determine the difference in the kinetic energy of a baseball thrown at speed v and one thrown at 2v and express the difference as a percentage [i.e., (K 2 K 1 )/K 1 100%]. Examples (a) 50% (b) 100% (c) 200% (d) 300% (e) 400%
9 Energy and W is defined as the amount of energy transferred to or from an object by means of a force. Examples W = F d = Fd cos(θ)
10 Energy and W is defined as the amount of energy transferred to or from an object by means of a force. Examples W = F d = Fd cos(θ) This is positive work; what would be negative?
11 Energy and The scalar product indicates that only the component of F parallel to d matters. Examples
12 Energy and The scalar product indicates that only the component of F parallel to d matters. Examples The component of the force perpendicular to the motion does no work.
13 Energy and If two or more forces act on the object, the net work is the sum of the individual works done by each force. Examples
14 Energy and If two or more forces act on the object, the net work is the sum of the individual works done by each force. Examples Remember: work can be zero or even negative.
15 Energy and -kinetic energy theorem: the change in kinetic energy of an object is equal to the net work done on that object. Examples K = W net
16 Energy and -kinetic energy theorem: the change in kinetic energy of an object is equal to the net work done on that object. Examples K = W net Positive work gives an increase in KE; negative work gives a decrease in KE.
17 Energy and Lecture Question 7.2 Two wooden blocks (masses m and 2m) are sliding with the same kinetic energy across a horizontal frictionless surface. The blocks then slide onto a rough horizontal surface. Let x A be the distance that the light block slides before coming to a stop and x B the distance that the heavy block slides before it stops. Then, Examples (a) x A = x B (b) x A = 2x B (c) x A = 4x B (d) x A = 0.5x B (e) x A = 0.25x B
18 Examples Energy and Like all forces, gravity can do positive or negative work on an object. Examples
19 Examples Energy and Like all forces, gravity can do positive or negative work on an object. Examples W g = mgd cos(θ)
20 Examples The force from a spring is given by F s = kx, where k is the spring constant (stiffness) and x is how far the spring is stretched/compressed. Energy and Examples Unstretched x = 0 Compressed x < 0 Sretched x > 0 x
21 Examples The force from a spring is given by F s = kx, where k is the spring constant (stiffness) and x is how far the spring is stretched/compressed. Energy and Examples Unstretched x = 0 Compressed x < 0 Sretched x > 0 x The force always points in the opposite direction of the displacement.
22 Examples Energy and To find the work done by a variable force, we compute the work done over a small distance many times and then add them up. Examples
23 Examples Energy and To find the work done by a variable force, we compute the work done over a small distance many times and then add them up. Examples Each small amount of work is W = F x or dw = Fdx.
24 Examples Energy and The work done by a variable force is written as an integral: W = F(x)dx W = F d r Examples
25 Examples Energy and The work done by a variable force is written as an integral: W = F(x)dx W = F d r Examples We compute the integral along a line of motion.
26 Examples Energy and The work done by a spring is therefore: W = x2 x 1 F(x)dx Examples
27 Examples Energy and The work done by a spring is therefore: W = = x2 x 1 x2 x 1 F(x)dx ( kx)dx Examples
28 Examples Energy and The work done by a spring is therefore: W = = x2 x 1 x2 x 1 F(x)dx ( kx)dx Examples = 1 2 kx2 x 2 x 1
29 Examples Energy and The work done by a spring is therefore: W = = x2 x 1 x2 x 1 F(x)dx ( kx)dx Examples = 1 2 kx2 x 2 x 1 = 1 2 k(x2 1 x 2 2)
30 Examples Energy and The work done by a spring is therefore: W = = x2 x 1 x2 x 1 F(x)dx ( kx)dx Examples = 1 2 kx2 x 2 x 1 = 1 2 k(x2 1 x 2 2) is positive if x 1 > x 2 (moves toward equilibrium) is negative if x 1 < x 2 (moves away from equilibrium)
31 Examples Energy and A general force: Examples F = F x î + F y ĵ + F zˆk
32 Examples Energy and A general force: Examples F = F x î + F y ĵ + F zˆk d r = dxî + dyĵ + dzˆk
33 Examples Energy and A general force: Examples F = F x î + F y ĵ + F zˆk d r = dxî + dyĵ + dzˆk W = r2 r 1 F d r
34 Examples Energy and A general force: Examples F = F x î + F y ĵ + F zˆk d r = dxî + dyĵ + dzˆk W = = r2 F d r r 1 x2 y2 F x dx + F y dy + x 1 y 1 z2 z 1 F z dz
35 Energy and is the rate of work done, defined as P = dw dt or P avg = W t Examples
36 Energy and is the rate of work done, defined as P = dw dt or P avg = W t Examples The unit of power is J/s which is known as the watt (W).
37 Energy and is the rate of work done, defined as P = dw dt or P avg = W t Examples The unit of power is J/s which is known as the watt (W). 1 horsepower = 746 watts
38 Energy and The instantaneous power is related to the velocity of an object: Examples P = dw dt = d[f cos(θ)x] dt
39 Energy and The instantaneous power is related to the velocity of an object: Examples P = dw d[f cos(θ)x] = dt dt = F cos(θ) dx dt
40 Energy and The instantaneous power is related to the velocity of an object: Examples P = dw d[f cos(θ)x] = dt dt = F cos(θ) dx dt P = Fv cos(θ)
41 Energy and The instantaneous power is related to the velocity of an object: Examples But in 3 dimensions, P = dw d[f cos(θ)x] = dt dt = F cos(θ) dx dt P = Fv cos(θ) P = F v
42 Energy and Lecture Question 7.4 A car is accelerated from rest to a speed v in a time interval t. Neglecting air resistance effects and assuming the engine is operating at its maximum power rating when accelerating, determine the time interval for the car to accelerate from rest to a speed 2v. (a) 2t (b) 4t (c) 2.5t (d) 3t (e) 3.5t Examples
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