#### Transcript Lecture_1 - Department of Mathematics

USSC2001 Energy Lecture 1 Potential and Kinetic Energy Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email [email protected] Tel (65) 6874-2749 QUESTIONS What falls faster – heavy or light objects ? hypotheses and supporting arguments experimental method Measurement what to measure how to measure HEIGHT Why is height a relative concept ? How does it depend on a reference position ? How does it depend on a reference length ? Does the distance that an object fall depend on the reference position ? Does it depend of the reference length ? Does A or B report a longer distance dropped ? A’s reference length = one stick B’s reference length = one stick LENGTH CONVERSION Question: If A’s stick has length = c times the length of B’s stick, how to convert from length in A-sticks into length in B-sticks ? Consider the equation length L = x (A-sticks) = y (B-sticks) Substitute the equation 1 (A-sticks) = c (B-sticks) to obtain x c (B-sticks) = y (B-sticks) Question: How is y related to x ? Why ? Question: How is x related to y ? Why ? HEIGHT CONVERSION Question: How are A’s and B’s height measurements related if they both use the same reference point (ie a point that has height zero) ? Question: What happens if A’s reference point is d (B-sticks) above B’s reference point ? Question: Let a point have height u (A-sticks) in A’s world and v (B-sticks) in B’s world. Derive an equation that expresses u in terms of v and another equation that expresses v in terms of u (both equations will also involve both c and d) Question: Graph u as a function of v and v as a function of u. What is the ‘geometry’ of c and d ? TUTORIAL 1 1. Two observers A and B measure the heights of a point x and a point y to be H_A(x) = 312, H_A(y) = 512 H_B(x) = 125, H_B(y) = 290 Compute constants (real numbers) r and c such that for any point z H_A(z) = r H_B(z) + c CLOCKS AND INVARIANCE How can we compare two time durations that start at different times? Our duration-length analogy might provide a clue! Clue: Find a repeatable physical process whose start and finish are points in time. Examples: Sandclock, Pendulum, Spring TUTORIAL 1 2. Three clocks A, B, C run at different (uniform) rates and have simultaneous readings, expressed in seconds, shown by the figure below? Express the readings of B and of C as functions of the reading of A. Hint: consider the previous tutorial problem 312 25 125 512 200 A 290 B 92 142 C THE MOTION OF A FALLING OBJECT Empirical measurements show that an object in a vacuum and starting with zero velocity falls a distance L = L(T) = gT^2 / 2 in a time interval having length T. Here g is a constant equal to 9.8 meters per squared second. Question: according to this formula does L depend on the position in time when or the position in space where the object starts to fall ? does it depend of the mass of the object ? Question: express height H = H(t) as a function of time t given H(t_0) and graph it THE SUPERPOSITION PRINCIPLE Empirical measurements show that the distance that an object in a vacuum drops satisfies the following superposition principle: the distance it drops if it starts with a nonzero velocity (positive for upwards and negative for downwards) is the sum of the distance that it would fall if it moved with constant velocity and the distance that it would fall if it’s initial velocity were equal to zero Question: express height H = H(t) as a function of time t given H(t_0) and V(t_0) and graph it TUTORIAL 1 3. A gun is fired directly at an object thrown directly upward when its height is maximum height. Show that the bullet will strikes the target unless it hits the ground before the object hits the ground (ignore air friction and assume that the ground is flat). Hint: the horizontal motion is superimposed on the vertical motion of the bullet fired from the gun. Also ignore the effects of wind and friction and Earth’s rotation. GRAPHICAL REPRESENTATION Consider a particle thrown upward from the ground h(t) t h(t 4 ) t2 t4 t t3 t1 We can use analytic geometry to construct a graph of this function using orthonormal planar coordinates AVERAGE AND INSTANTANEOUS VELOCITIES The average velocity over h ( t ) the interval [ t 3 , t 4 ] is the slope of the dotted line h(t4 ) h(t3 ) Vav t t 4 Tangent Secant 3 The instantaneous velocity at t t 4is the slope of the solid line and equals the derivative dh/dt evaluated at t t 4 t2 t1 t4 t3 t COMPUTING INSTANTANEOUS VELOCITIES v( t ) t2 t1 t4 t t3 The instantaneous velocity v(t) is computed using differential calculus h ( t 4 ) h ( t 4 ) dh v( t 4 ) | t t lim 4 dt 0 2 1 If h ( t ) v ( 0 ) t 2 gt the graph of v is shown above TUTORIAL 1 4. Compute the velocity V(t) for a falling object given H(t_0) and V(t_0). Show that the quantity M V(t)^2 / 2 + M gH(t) does not change with time. The first term is called kinetic energy and the second is called (gravitational) potential energy. SCALES AND MASS MEASUREMENTS Masses of objects can be compared using a scale. A B Question: in which direction will the right side move ? Answer: up, down iff mass B < , > mass A TUTORIAL 1 5. Discuss how the sum of the potential energy of two masses at opposite sides of a balanced board of a fulcrum changes as the board is very slowly moved. COMPUTING DISTANCE FROM VELOCITY v( t ) t t The fundamental theorem of calculus says that h equals the signed area under the graph of v h(t) s t dh ( u ) s 0 du | u s ds st s 0 v ( s ) ds SYSTEMS OF UNITS SI (International System) of seven base units include: meter (m)– length or distance traveled by light in a vacuum during 1/299792458 of a second second (s) – time taken by 9192631770 oscillations of the light (of a specified frequency) emitted by a cesium-133 atom kilogram (kg) – mass of a certain platinum-iridium cylinder These base units can be used to define derived units 2 newton (N) – force unit = (1 kg ) (1 m / s ) joule (J) - energy unit = (1 N) x (1 m)