Physics 215 – Fall 2014Lecture Welcome back to Physics 215 Today’s agenda: More gravitational potential energy Potential energy of a spring Work-kinetic energy theorem

Physics 215 – Fall 2014Lecture Current homework assignment HW7: –Knight Textbook Ch.9: 54, 72 –Ch.10: 48, 68, 76 –Ch.11: 50, 64 –Due Wednesday, Oct. 22 nd in recitation

Physics 215 – Fall 2014Lecture Gravitational Potential Energy For an object of mass m near the surface of the earth: U g = mgh h is height above arbitrary reference line Measured in Joules -- J (like kinetic energy)

Physics 215 – Fall 2014Lecture Total energy for object moving under gravity  E = U g + K = constant  * E is called the (mechanical) energy  * It is conserved: (½) mv 2 + mgh = constant

Physics 215 – Fall 2014Lecture A ball of mass m=7 kg attached to a massless string of length R=3 m is released from the position shown in the figure below. (a) Find magnitude of velocity of the ball at the lowest point on its path. (b) Find the tension in the string at that point.

Physics 215 – Fall 2014Lecture Stopped-pendulum demo Pendulum swings to same height on other side of vertical What if pendulum string is impeded ~1/2- way along its length? Will height on other side of vertical be: 1.Greater than original height 2.Same as original height 3.Less than original height?

Physics 215 – Fall 2014Lecture A block is released from rest on a frictionless incline. The block travels to the bottom of the left incline and then moves up the right incline which is steeper than the left side.

Physics 215 – Fall 2014Lecture Springs -- Elastic potential energy Force F = -kx (Hooke’s law) Area of triangle lying under straight line graph of F vs. x = (1/2)(+/-x)(-/+kx) F x F = -kx U s = (1/2)kx 2 frictionless table

Physics 215 – Fall 2014Lecture (Horizontal) Spring frictionless table (1/2)kx 2 + (1/2)mv 2 = constant x = displacement from relaxed state of spring Elastic potential energy stored in spring: U s = (1/2)kx 2

Physics 215 – Fall 2014Lecture A 0.5 kg mass is attached to a spring on a horizontal frictionless table. The mass is pulled to stretch the spring 5.0 cm and is released from rest. When the mass crosses the point at which the spring is not stretched, x = 0, its speed is 20 cm/s. If the experiment is repeated with a 10.0 cm initial stretch, what speed will the mass have when it crosses x = 0 ? 1.40 cm/s 2.0 cm/s 3.20 cm/s 4.10 cm/s

Physics 215 – Fall 2014Lecture Mass hanging on spring Now oscillations are about equilibrium point of spring + mass Otherwise, motion is same as horizontal mass + spring on frictionless table (1/2)mv 2 = (1/2)ka 2 - (1/2)kz 2

Physics 215 – Fall 2014Lecture Work, Energy Newton’s Laws are vector equations Sometimes easier to relate speed of a particle to how far it moves under a force – a single equation can be used – need to introduce concept of work

Physics 215 – Fall 2014Lecture What is work? Assume constant force in 1D Consider: v F 2 = v I 2 + 2a s Multiply by m/2  (1/2)mv F 2 - (1/2)mv I 2 = m a s But: F = ma  (1/2)mv F 2 - (1/2)mv I 2 = F s

Physics 215 – Fall 2014Lecture Work-Kinetic Energy theorem (1) (1/2)mv F 2 - (1/2)mv I 2 = F s Points: W = Fs  defines work done on particle = force times displacement K = (1/2)mv 2  defines kinetic energy =1/2 mass times square of v

Physics 215 – Fall 2014Lecture Work-Kinetic Energy demo Cart, force probe, and motion detector Plot v 2 vs. x – gradient 2F/m constant F (measure) -- pulling with string Weigh cart and masses in advance

Physics 215 – Fall 2014Lecture Conclusions from experiment Although the motion of the two carts looks very different (i.e., different amounts of time, accelerations, and final speeds), there is a quantity that is the same for both at the end of the motion. It is (1/2) mv 2 and is called the (final) kinetic energy of the carts. Moreover, this quantity happens to have the same value as F  s, which is given the name work.

Physics 215 – Fall 2014Lecture Improved definition of work For forces, write F  F AB Thus W = F s  W AB = F AB  s A is work done on A by B as A undergoes displacement  s A Work-kinetic energy theorem: W net,A =  B W AB =  K

Physics 215 – Fall 2014Lecture Non-constant force … x F xx F(x) Work done in small interval  x  W = F  x Total W done from A to B     F  x = Area under curve! B A

Physics 215 – Fall 2014Lecture The net work done on an object is equal to the change in kinetic energy of the object. The Work - Kinetic Energy Theorem W net =  K = K f - K i

Physics 215 – Fall 2014Lecture It takes equal distances to stop each ball. 2.It takes equal time intervals to stop each ball. 3.Both of the above. 4.Neither of the above. Suppose a tennis ball and a bowling ball are rolling toward you. The tennis ball is moving much faster, but both have the same momentum (mv), and you exert the same force to stop each. Which of the following statements is correct?

Physics 215 – Fall 2014Lecture less than. 2.equal to 3.greater than Suppose a tennis ball and a bowling ball are rolling toward you. Both have the same momentum (mv), and you exert the same force to stop each. It takes equal time intervals to stop each ball. The distance taken for the bowling ball to stop is the distance taken for the tennis ball to stop.

Physics 215 – Fall 2014Lecture Two carts of different mass are accelerated from rest on a low-friction track by the same force for the same time interval. Cart B has greater mass than cart A (m B > m A ). The final speed of cart A is greater than that of cart B (v A > v B ). After the force has stopped acting on the carts, the kinetic energy of cart B is 1.less than the kinetic energy of cart A (K B < K A ). 2.equal to the kinetic energy of cart A (K B = K A ). 3.greater than the kinetic energy of cart A (K B > K A ). 4.“Can’t tell.”

Physics 215 – Fall 2014Lecture Kinetic energy of an object: Work done on object 1 by object 2: Revised definitions for Work and Kinetic Energy

Physics 215 – Fall 2014Lecture Scalar (or “dot”) product of vectors The scalar product is a way to combine two vectors to obtain a number (or scalar). It is indicated by a dot () between the two vectors. (Note: component of A in direction n is just An)

Physics 215 – Fall 2014Lecture positive 2.negative 3.equal to zero 4.“Can’t tell.” Is the scalar (“dot”) product of the two vectors

Physics 215 – Fall 2014Lecture positive 2.negative 3.equal to zero 4.“Can’t tell.” Is the scalar (“dot”) product of the two vectors

Physics 215 – Fall 2014Lecture Two identical blocks slide down two frictionless ramps. Both blocks start from the same height, but block A is on a steeper incline than block B. The speed of block A at the bottom of its ramp is 1.less than the speed of block B. 2.equal to the speed of block B. 3.greater than the speed of block B. 4.“Can’t tell.”

Physics 215 – Fall 2014Lecture Solution Which forces do work on block? Which, if any, are constant? What is F  s for motion?

Physics 215 – Fall 2014Lecture Work done by gravity N ss mgmg Work W = -mg j  s Therefore, W = -mg  h N does no work! i j

Physics 215 – Fall 2014Lecture Reading assignment More W-KE Theorem Conservative and non-conservative forces Power Finish chapter 11 in textbook