Physics 101: Lecture 13 Rotational Kinetic Energy and Inertia Exam II Physics 101: Lecture 13 Rotational Kinetic Energy and Inertia 1

Today’s Quizzes 1. A 2-kg stone is thrown vertically down with an initial velocity 12m/s from the tower of Pisa from a height of 30 m. Assume that the potential energy at the ground is zero. Neglect air resistance. 1) What is the potential energy of the stone before falling (relative to the ground)? [5] 2) What is the speed of the stone just before hitting the ground? 2. A car does 45,000 J of work to travel at constant speed for 1.2 km along a horizontal road. 1) What is the average retarding force acting on the car? 2) The mass of the car is 1500 kg and the speed is 15 m/s. How much distance do you expect that the car can traverse after turning off its engine? ------------------------------------------------------------------------------------------------------------------------ 1. A 50 kg skier starts with an initial speed of 7m/s from the top of an incline. The incline has a slope of 20 degrees. Assume friction can be neglected. The total height of the hill is 120 m. 1) What is the kinetic energy of the skier at the bottom of the incline? 2)After reaching the bottom of the incline, the skier goes onto a horizontal surface with friction.  If the coefficient of kinetic friction between the skier and the ground is 0.3, what is the distance she can traverse before she completely stops? 2. John (his mass = 70 kg) and Mary (her mass = 55 kg) start at the same time from the top of two water-slides 10 m above the water level. John’s slide has an incline of 35 degrees while Mary’s slide has an incline of 15 degrees. Neglect friction. What is John’s speed when he reaches the water? What is Mary’s speed when she reaches the water? Who gets to the water first? (Do not calculate!)

Yesterday I… A) Watched the parade B) Watched fireworks C) Prepared for Exam 2 D) Both A and B E) All of the above

Center of Mass Some objects can’t be balanced on a single point Center of Mass = Balance point Some objects can’t be balanced on a single point 46

Example: center of mass m = 0.140 kg 0.1m M = 0.515

Summary Collisions and Explosions Center of Mass (Balance Point) Draw “before”, “after” Define system so that Fext = 0 Set up axes Compute Ptotal “before” Compute Ptotal “after” Set them equal to each other Center of Mass (Balance Point) 50

Rotational Inertia, I Tells how difficult it is get object spinning. Just like mass tells you how difficult it is to get object moving. Fnet= m a Linear Motion τnet = I α Rotational Motion I = S miri2 (units kg m2) Note! Rotational Inertia depends on what you are spinning about (basically the ri in the equation). 13

Inertia Rods Two batons have equal mass and length. Which will be “easier” to spin A) Mass on ends B) Same C) Mass in center I = S m r2 Further mass is from axis of rotation, greater moment of inertia (harder to spin) 21

Example: baseball bat

Rotational Inertia Table For objects with finite number of masses, use I = S m r2. For “continuous” objects, use table below. 33

Example: Rolling An hoop with mass M, radius R, and moment of inertia I = MR2 rolls without slipping down a plane inclined at an angle  = 30o with respect to horizontal. What is its acceleration? Consider CM motion and rotation about the CM separately when solving this problem I M R  29

Rolling... Static friction f causes rolling. It is an unknown, so we must solve for it. First consider the free body diagram of the object and use In the x direction Now consider rotation about the CM and use  = I M y x f Mg  R  33

Rolling... We have two equations: We can combine these to eliminate f:  36

What will change the acceleration of an object rolling down an incline plane? A. The mass of the object B. The radius of the object C. The type of object (solid, hollow, sphere, disk) D. The angle of the incline. E. C and D only

Rotational Energy It is moving so it is a type of Kinetic Energy (go back and rename the first) Rotaional KE Translational KE

Example: cylinder rolling Consider a cylinder with radius R and mass M, rolling w/o slipping down a ramp. Determine the ratio of the translational to rotational KE. Friction causes object to roll, but if it rolls w/o slipping friction does NO work! W = F d cos q d is zero for point in contact No dissipated work, energy is conserved Need to include both translation and rotation kinetic energy. H Finish up 43

Example: cylinder rolling Consider a cylinder with radius R and mass M, rolling w/o slipping down a ramp. Determine the ratio of the translational to rotational KE. Translational: Rotational: use and Ratio: Finish up H 43

Example: cylinder rolling What is the velocity of the cylinder at the bottom of the ramp? H 45