* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Example - mrdsample
Equations of motion wikipedia , lookup
Internal energy wikipedia , lookup
Laplace–Runge–Lenz vector wikipedia , lookup
Faster-than-light wikipedia , lookup
Atomic theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Fictitious force wikipedia , lookup
Hunting oscillation wikipedia , lookup
Hooke's law wikipedia , lookup
Relativistic angular momentum wikipedia , lookup
Classical mechanics wikipedia , lookup
Matter wave wikipedia , lookup
Seismometer wikipedia , lookup
Mass in special relativity wikipedia , lookup
Electromagnetic mass wikipedia , lookup
Specific impulse wikipedia , lookup
Rigid body dynamics wikipedia , lookup
Work (thermodynamics) wikipedia , lookup
Center of mass wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Centripetal force wikipedia , lookup
AP C UNIT 3 WORK & ENERGY SCALAR PRODUCT or DOT PRODUCT Dot Product is defined as the magnitude of 1st ( a ) times scalar component of 2nd vector ( b ) along direction of 1st. Essentially, the dot product gives you information about how much of each vector lies along the direction of the other. a b ab cos it’s a scalar result where Ф is the angle between a and b. a b THIS IS NOT THE SAME AS ADDING 2 VECTORS YIELDING A RESULTANT The reason the dot product is used in physics is because the operation between certain vector quantities produce meaningful physical answers such as WORK. Since the dot product involves the cosine then IF Ф = 90o THEN PRODUCT = 0 IF Ф = 0o THEN PRODUCT = MAXIMUM This is consistent in that when vectors are perpendicular neither lies along the other, therefore an answer of zero results. Directional properties of the Dot Product include: i i j j k k 1 and i j j k i k 0 Unit vector form of dot product: a b (a x i a y j a z k ) (bx i by j bz k ) if you distribute, this would reduce to… a b axbx a y by az bz *Note that there is no direction associated with result but answer can be negative depending on angle. Calculate the dot product of the following vectors and find the angle between them: A = -3i + 5j B = 6i +14j More Calculus - Derivative of a Product: When taking the derivative of two functions multiplied together, the derivative is: The 1st function times the derivative of the 2nd plus the derivative of the 1st times the 2nd function. d dv du uv u v dx dx dx u v v u ' ' Example…find dy/dx y (3x 1)( x 2) 2 u v y (3x 2 1)( x 2) dy (3 x 2 1) 1 ( x 2) 6 x dx dy 2 9 x 12 x 1 dx *Could have ‘foiled’ and then performed power rule as well Chain Rule: The chain rule is used when there is a function within a function. f‘ (x) = f‘( g(x) ) (g'(x)) f (x) = f ( g(x) ) Think of the functions f and g as ``layers'' of a problem. Function f is the ``outer layer'' and function g is the ``inner layer.'' Thus, the chain rule tells us to first differentiate the outer layer, leaving the inner layer unchanged (the term f'(g(x))) , then differentiate the inner layer (the term g'(x) ). f ( x) ( x 5) 3 The inner layer, g(x) is 3 1 2 f 2( x 5) 3x 2 (x3 + 5) The outer layer f(x) is (x)2 Derivative of outer times derivative of inner with respect to x f 6 x 30 x 5 2 Example: Find the derivative of f’ function of a function 1 f ( x) 2 x 3 Work Done by a Constant Force Work is defined as an external force (F) moving through a displacement (Δr). How much force lies along the movement of an object. Positive & Negative Work In all 4 cases, the force has the same magnitude and the displacement of the object is to the right with the same magnitude. Rank the situations from most positive to most negative. A crate of mass, M, is dragged along a level rough surface a distance, x, by a force, F as shown. The coefficient of friction is uk. Find the net work done on the crate in terms of given variables and constants. Work done by a varying force x2 W F ( x)dx x1 If F(x) = 4x2 then find the work done on a particle that moves from x = 1m to x = 5m. Example: Suppose a mass moves with a trajectory defined by the position vector r (t ) (te t 10 2 ˆ ˆ )i (t 3t ) j Find the work done by the force, F 10iˆ 4 ˆj over the interval from t = 1 to t = 2. Work-Energy Theorem x2 W F ( x)dx x1 A force, F(x) = 2-4x, acts on a 7.0kg mass. What is the final speed of the mass as it is moved from x=5m to x=2? Assume mass starts from rest at t=0. Hooke’s Law Work done by Spring Negative means that the force opposes the displacement from equilibrium If block is pulled to right, the force by spring is NOT constant via Hooke’s Law. Therefore, the work done by spring must use avg force or be integrated as: Fs A plot of spring force vs displacement reveals a slope equal to spring constant, k k1 k1 kk 2 k1 2 k2 Two springs are attached to a block in series and parallel as shown above. Determine the effective spring constant for each situation in terms of k1 and k2 . What would spring constant be if a mass was attached to a massless spring that stretched a distance x? What if same mass was attached to 2 springs as shown? How would stretch, x, differ for each spring? Power Rate at which work is done or energy is transferred A 4kg particle moves along the x-axis. Its position varies with time according to x = t + 2t3, where x is in meters and t is in seconds. Find the the power being delivered to the particle at any time t OR If a projectile thrown directly upward reaches a maximum height h and spends a total time in the air T, the average power of the gravitational force during the trajectory is: a) 2mgh / T b) -2mgh / T c) 0 d) mgh / T e) -mgh / T Potential Energy As height above Earth increases… Conservative & Non-conservative Forces The work a conservative force does on an object in moving it from A to B is path independent - it depends only on the end points of the motion. Force of gravity and the spring force are conservative forces. Conservative forces ‘store’ energy…available for kinetic energy The work done by non-conservative (or dissipative) forces in going from A to B depends on the path taken. Friction is non-conservative. Nonconservative forces don’t ‘store’ energy. CONSERVATION OF NRG Total energy in a closed system remains unchanged Work done by NC forces or friction is positive in the above formula. No need to put in minus sign. Work done by friction occurs on left side as minus but becomes + when taken to other side. Energy worksheet Potential Energy Function Diagrams Us (x) vs Restoring Force Relationship f o r c e position position Potential Energy and Conservative Restoring Forces (gravity, spring) The instantaneous restoring force is equal to the negative derivative of the potential energy function. In the case of a mass oscillating on a horizontal frictionless surface we can verify this relationship: Example1 A certain spring is found not to obey Hooke’s Law; it exerts a restoring force F(x) = -60x-18x2. a) Calculate the potential energy function U(x) for this spring. Let U = 0 when x = 0. b) An object with mass 0.90kg on a frictionless, horizontal surface is attached to this spring, pulled a distance 1.00m to the right to stretch the spring and released. What is the speed of the object when it is 0.50m to the right of x=0? Potential energy diagram states of equilibrium: Points of equilibrium are where the force is zero (slope = zero). x3 and x5 are points of stable equilibrium or energy wells. If the system is slightly displaced to either side the forces on either side will return the object back to these positions. x6 is a position of neutral equilibrium. Since there is no net force acting on the object (slope of U(x) = 0) it must either possess only potential energy and be at rest or, it also possesses kinetic energy and must be moving at a constant velocity. x4 is a position of unstable equilibrium. If the object is displaced ever so slightly from this position, the internal forces on either side will act to encourage further displacement instead of returning it back to x4. Example2 A 5-kg mass moving along the xaxis passes through the origin with an initial velocity of 3m/s. Its potential energy as a function of its position is given in the graph. a) How much total energy does the mass have as it passes through the origin? b) Between 2.5m and 5m, is the mass gaining speed or losing speed? c) How fast is it moving at 7.5m? d) How much potential energy would have to be present for the mass to stop moving? Turning points Positions where potential energy equals the total mechanical energy, Umax= E, are called turning points A particle moves along the x-axis according to the following potential energy function: 2.4 Nm U ( x) (0.60 N ) x x 2 Find the positions of equilibrium for the particle. Find F(x) when ax U ( x) 2 2 b x CENTER OF MASS Center of Mass (COM) COM = average position of mass of an object. A point where all of the mass could be considered to be located. Similarly we can show that: Example: A rectangular wire frame lies in the xy plane. If the linear mass density of the wires is λ, except for the bottom segment which has twice the density, where is the center of mass of the wire frame? a )( a / 2, b / 2) a 2a 2b b b) , 2 3a 2b 2 a 2b c) , 2 3 a b 2a 2b d) , 2 2 3a 2b a b 3a 2b e) , 2 2 2a 2b (a, b) Example A 40kg woman and a 55kg man stand at the left and right ends of a 3.0m long plank. The plank is floating in water. Ignore frictional effects between the water and the plank. The mass of the plank is 15.0kg. The woman is intrigued by the amazing man and walks over to him on the right side. How far does the plank move? • C.O.M. acts like mass of system…it can have vcom and acom • C.O.M. remains constant through problem • Ie; explosion: com of fragments would follow parabola Linear Momentum F ma Momentum is a vector Units = kgm/s Impulse Impulse equals the change in momentum and is also equal to the area under a force-time curve given by: Units = Ns Impulse is a vector Avg Force: Same area that is under Ft curve. Favg Example1 Momentum of object moving along x-axis is given by p(t) = 2t2 -5t + 3. a) Find net force at t = 0.50s b) Find net force when object first reverses original direction. c) Find net impulse between 0 and 0.80s. Example2: A batter strikes a ball (m =0.145kg) with force given by F = (1.6x107 t – 6.0x109 t2 )i between 0 and 2.5ms. At t =0, v = -(40i + 5j). a) Find impulse from bat on ball from t= 0 to t = 2.5ms. b) Find the impulse of gravity on ball for same time. c) Find avg force on ball by bat d) Find momentum & velocity of ball at t = 2.5ms in unit vector notation. Conservation of p Internal forces External forces Types of Collisions: ELASTIC: Both momentum & KE are both conserved. Whatever KE is spent deforming objects is recovered after objects separate. INELASTIC: KE is not conserved (some is used to produce sound, heat, etc…nonconservative processes) TOTALLY INELASTIC: Same as above, but objects remain stuck together after collision. Example 1: A block of mass M is resting on a horizontal, frictionless table and is attached as shown to a relaxed spring of spring constant k. A second block of mass 2M and initial speed vo collides with and sticks to the first block. Develop expressions for the following quantities in terms of M, k, & vo. A) Find v, the speed of the blocks immediately after impact. B) x, the maximum distance the spring is compressed EXAMPLE 2: A track consists of a frictionless arc XY, which is a quarter-circle of radius R, and a rough horizontal section YZ. Block A of mass M is released from rest at point X, slides down the curved section of the track, and collides instantaneously and inelastically with identical block B at point Y. The two blocks move together to the right, sliding past point P, which is a distance l from point Y. The coefficient of kinetic friction between the blocks and the horizontal part of the track is . Express your answers in terms of M, l, , R, & g. A) Determine the speed of the combined blocks immediately after the collision. B) Determine the total KE lost from point X to point P. Example 3: 2 identical masses are connected by a spring (k) resting on a frictionless horizontal surface. The right mass is initially in contact with the wall. A brief blow to the left block leaves it with an initial velocity vo to the right. After the spring is maximally compressed, it moves to the left, away from the wall continuing to oscillate. a) What is the net momentum of the two masses after they leave the wall? b) What is the maximum compression of the spring after the two masses have left the wall? Example4 A railroad handcar is moving along straight, frictionless tracks with negligible air resistance. In the following cases, the car initially has a total mass (car + contents) of 200kg and is traveling at 5.00m/s, east. Find the final velocity of the car in each case: a) A 25.0kg mass is thrown sideways out of the car with a velocity of 2.00m/s relative to the car’s initial velocity. b) A 25.0kg mass is thrown backward out of the car with a velocity of 5.00m/s relative to the initial motion of the car. c) A 25.0kg mass is thrown into the car with a velocity of 6.00m/s relative to the ground and opposite in direction to the initial velocity of the car. A massive frog drops vertically from a tree branch onto a skateboard that moves horizontally below. When the frog lands, the skateboard slows, consistent with the conservation of momentum. The impulse that slows the skateboard is a) The friction force of the frog’s feet acting backward on the skateboard x time during which the speed changes b) equal and opposite to the impulse that brings the frog up to speed c) both of these d) neither of these 2D Collisions: Momentum must be conserved in both x and y directions. Example: A car with mass 1000kg is moving north at 15m/s. At an intersection, it collides with an SUV with mass 2000kg, traveling east at 10m/s. Both vehicles become entangled and move off together after the collision. You can ignore any external forces. a) Determine the velocity of the wreckage immediately after the collision. b) Determine the % of KE lost during the collision. Example2 Spheres A (0.020kg), B (0.030kg), and C (0.050kg) are each approaching the origin as they slide across a frictionless air table. The initial velocities of A and B are given. All 3 spheres arrive at the same time at origin and stick together. vB = 0.50m/s 60o vA=1.50m/s C a)What must be the velocity of C if all three objects end up moving at 0.50m/s east after the collision? b) Find the change in KE of the system as a result of the collision Example3 An elastic collision occurs between 2 pucks on a frictionless air hockey table. Puck A has initial speed of 4.00m/s east (mass 0.500kg) and final speed of 2.00m/s in an unknown direction. Puck B has a mass of 0.300kg and is initially at rest. Find the final speed of B and the unknown angles. A α β B Example: A mass m sits at top of smooth semicircular cutout of larger mass M. M sits on a frictionless surface. m R M a) When m is released from rest, describe the motion of the 2 masses. b) When the mass is at the bottom of the circle, what is its speed with respect to the table? c) If M were fixed to the table, would this increase or decrease the maximum speed of m? d) When m has reached the top left side of the cutout, how far has M been displaced horizontally? Example A projectile is fired from the edge of a cliff 100m high with initial speed of 60m/s at an angle of 45o. A) Calculate the x-y location and velocity coordinates of the particle at t = 5s. Suppose the projectile experiences an internal explosion at t=4s with an internal force purely in the y-direction, causing it to break into a 2kg and 1kg fragment. B) If the 2kg fragment is 77m above the height of the cliff at t = 5s, what is the y position of the 1kg piece? C) If the speed of the 2kg fragment is 46m/s and the fragment is falling at t=5s, what is the y component of the velocity of the 1kg fragment?