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PHYSICS 149: Lecture 15 • Chapter 6: Conservation of Energy – 6.3 Kinetic Energy – 6.4 Gravitational Potential Energy Lecture 15 Purdue University, Physics 149 1 ILQ 1 Mimas orbits Saturn at a distance D. Enceladus orbits Saturn at a distance 4D. What is the ratio of the periods of their orbits? A) Tm/Te = 1/8 T23∝ r 2 ⎛⎞ 3 B) Tm/Te =1/4= 1/4 Tm ⎛⎞D ⎜⎟= ⎜⎟ ⎝⎠TDE ⎝⎠4 C) Tm/Te = 1/2 ⎛⎞T 11 D) T /T =2= 2 ⎜⎟m = = m e ⎝⎠TE 64 8 Lecture 15 Purdue University, Physics 149 2 ILQ 2 A pendulum bob swings back and forth along a circular path. Does the tension in the string do any work on the bob? Does gravity do work on the bob? A) only tension does work B) both do work C) neither do work D) only gravity does work Lecture 15 Purdue University, Physics 149 3 Energy • Energy is “conserved” meaning it can not be created nor destroyed – Can change form – Can be transferred • Total Energy of an isolated system does not change with time • Forms – Kinetic Energy Motion – Potential Energy Stored – Heat – Mass (E=mc2) • Units: Joules = kg m2 /s/ s2 Lecture 15 Purdue University, Physics 149 4 Definition of “Work” in Physics • Work is a scalar quantity (not a vector quantity). • Units: J (Joule), N⋅m, kg⋅m2/s2, etc. – Unit conversion: 1 J = 1N1 N⋅m = 1kg1 kg⋅m2/s2 • Work is denoted by W (not to be confused by weight W). Lecture 15 Purdue University, Physics 149 5 Total Work • When several forces act on an object, the “total” work is the sum of the work done by each force individually: Lecture 15 Purdue University, Physics 149 6 ILQ 1 You are towing a car up a hill with constant velocity. The work done on the car by the normal force is: A) positive B) negative C) zero FN V T Normal force is perpendicular to direction of W disp lacemen t, so work is zero. Lecture 15 Purdue University, Physics 149 7 ILQ 2 You are towing a car up a hill with constant velocity. The work done on the car by the gravitational force is: A) positive B) negative C) zero FN V T Gravity is pushing against the direction of W motion so it is negative. Lecture 15 Purdue University, Physics 149 8 ILQ 3 You are towing a car up a hill with constant velocity. The work done on the car by the tension force is: A) positive B) negative C) zero FN V T The force of tension is in the same direction as W the motion of the car, making the work positive. Lecture 15 Purdue University, Physics 149 9 ILQ 4 You are towing a car up a hill with constant velocity. The total work done on the car by all forces is: A) positive B) negative C) zero FN V T The total work done is positive because the car is moving up the hill. (Not quite!) W 2 2 W=KEf-KEi=(0.5mvf ) - (0.5mvi ). Because the final and initial velocities are the same, there is no change in kinetic energy, and therefore no total work is done. Lecture 15 Purdue University, Physics 149 10 Problem A box is pulled up a rough (μ > 0) incline by a rope- pulley-weight arrangement as shown below. How many forces are doing (non-zero) work on the box? A) 0 B) 1 C) 2 D) 3 E) 4 Lecture 15 Purdue University, Physics 149 11 Solution Draw FBD of box: N v T z Consider direction of motion of the box z Any force not perpendicular to the motion will do work: f N does no work (perp. to v) T does positive work mg f does negative work 3f3 forces do work mg does negative work Lecture 15 Purdue University, Physics 149 12 Example: Block with Friction • A block is sliding on a surface with an initial speed of 5 m/s. If the coefficient of kinetic friction between the block and table is 0.4, how far does the block travel before stopping? y N y-direction: FFma=ma f N-mg = 0 x N = mg mg Work W = Δ K 2 2 WN = 0 -μmg Δx = ½ m (vf –v0 ) 2 Wmg = 0 -μg Δx=½x = ½ (0 – v0 ) 2 Wf = f Δx cos(180) μg Δx = ½ v0 2 = -μmg Δx Δx = ½ v0 / μg 5 m/s = 313.1 meters Lecture 15 Purdue University, Physics 149 13 Kinetic Energy: Motion • Apply constant force along x-direction to a point particle m. W = Fx Δx 1 = m ax Δx 2 2 recall:axΔx = (vx − vx0 ) 2 2 = ½ m (vf –v0 ) 2 • Work changes ½ m v2 • Define Kinetic Energy K=½mvK = ½ m v2 WkWork-Kine tic Energy W = Δ K For Point Particles Theorem Lecture 15 Purdue University, Physics 149 14 Translational Kinetic Energy • When an object of mass m is moving with speed v (the maggy),jnitude of instantaneous velocity), the object’s “translational kinetic energy” is defined as follows: • Kinetic energy is a scalar quantity. • Units: J, N ⋅m, kg ⋅m2/s2, etc. • Kinetic energy is denoted by K. • Translational kinetic energy means the total work done on the object to accelerate it to that speed starting from rest. • Translational kinetic energy is often called the “kinetic energy” if it is clearly distinguished from the rotational energy or internal energy. Lecture 15 Purdue University, Physics 149 15 Work - Kinetic Energy Theorem 1 1 = K − K = mv2 − mv2 f i 2 f 2 i • The work done on an object by the “net” force (whether the net force is constant or variable) is equal to the change in the kinetic energy. • Or, the “total” work done on the object is equal to the change in the kinetic energy. Lecture 15 Purdue University, Physics 149 16 ILQ Compare the kinetic energy of two balls: Ball 1: mass m thrown with speed 2v Ball 2: mass 2m thrown with speed v A) K1 = 4K2 B) K1 = K2 C) 2K1 = K2 D) K1 = 2K2 Lecture 15 Purdue University, Physics 149 17 Work Done by Gravity 1 •Examppple 1: Drop ball Y = h Wg = (mg)(s)cosθ i s = h S 0 Wg = mghcos(0 ) = mgh mg y Δy = yf-yi = -h Yf = 0 x Wg = -mgΔy Lecture 15 Purdue University, Physics 149 18 Work Done by Gravity 2 •Examppple 2: Toss ball up Y = h Wg = (mg)(s)cosθ i s = h S W = mghcos(1800) = -mgh mg g y Δy = yf-yi = +h Y = 0 f x Wg = -mgΔy Lecture 15 Purdue University, Physics 149 19 Work Done by Gravity 3 •Example 3: Slide block down incline Wg = (mg)(s)cosθ s = h/cosθ h θ Wg = mg(h/cosθ)cosθ mg S Wg = mgh Δy = yf-yi = -h Wg = -mgΔy Lecture 15 Purdue University, Physics 149 20 Work Done by Gravity • Depends only on initial and final height! • Wg = -mg(yf - yi)=) = -mgΔy – Independent of path – If you end up where you began, Wg =0= 0 – Note: can do work “against” gravity, then get gravity to do work back. – Define: Potential Energy…… We call this a “Conservative Force” because we can define a “Potential Energy ” to go with it. Lecture 15 Purdue University, Physics 149 21 Work Done by Gravity • Question: Does the work done by gravity depend on the path taken? θ mg Δr Left Case: Wgrav = F⋅Δr⋅cosθ = mg⋅|Δy|⋅cos0° = mg|Δy| Middle Case: Wgrav = F⋅Δr⋅cosθ = mg|Δy| (because Δr⋅cosθ = |Δy|) Right Case: Wgrav = mg|Δy| (because each segment can be treated like the middle case) • Answer: The work done by gravity is independent of path–that is, the work depends only on the initial and final positions (|Δy|). • This kind of force is called “conservative force.” Lecture 15 Purdue University, Physics 149 22 Potential (stored) Energy • “Stored” gravitational energy can be converted to kinetic energy -m g Δy= ΔK 0 = ΔK + m g Δy define Ug = mgy 0 = ΔK + ΔUg W = ΔK ΔU = -WC • Works for any CONSERVATIVE force Gravity Ug = m g y 2 Spring Us = 1/2 k x NOT friction Lecture 15 Purdue University, Physics 149 23 Potential Only change in potential energy is important Lecture 15 Purdue University, Physics 149 24 What is Potential Energy? • An object is thrown up vertically, and it reaches top. Assume no air resistance. The work done by gravity (near the surface of Earth) is W =F= F⋅Δr⋅cosθ =mg= mg⋅Δy⋅cos180° = –mgΔy grav vf=0 In this problem, gravity is the only force. Thus, Wttltotal = Wgrav = –mgΔy According to work-energy theorem, 2 2 Wtotal = –mgΔy = –½mvi (because ½mvf =0) 2 vi That is, the initial kinetic energy (Ki = ½mvi ) has been “stored” in (or transformed into) the form of mgΔy at the top. And, it has the “potential” to do work (or to become kinetic energy). • Stored energy due to the interaction of an object with something else (in this case, gravity) that can easily be recovered as kinetic energy is called potential energy. Lecture 15 Purdue University, Physics 149 25 Definition of Potential Energy • The change in potential energy is equal to the negative of the work done by the conservative forces. – Potential energy can be defined only for the conservative forces. For the non-conservative forces, potential energy can not be defined in the first place. – There is no way to calculate the absolute value of the potential energy. Only the change in potential energy is important. – The choice of the zero point of potential energy is arbitrary.
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