Physics 151: Lecture 11, Pg 1 Physics 151: Lecture 11 l Homework #4 (10/2/06, 5 PM) : Problems from Ch.6 on the webassign l Topics çReview: Forces and uniform circular motion - Ch 6.1 çForces and non-uniform circular motion - Ch 6.2 çAccelerated reference frames - Ch 6.3

Physics 151: Lecture 11, Pg 2 Lecture 11, ACT 1 A split highway has a number of lanes for traffic. For traffic going in one direction, the radius for the inside of the curve is half the radius for the outside. One car, car A, travels on the inside while another car of equal mass, car B, travels at equal speed on the outside of the curve. Which statement about resultant forces on the cars is correct ? a. The force on A is half the force on B. b. The force on B is half the force on A. c. The force on A is four times the force on B. d. The force on B is four times the force on A. e.There is no net resultant force on either as long as they stay on the road while turning.

Physics 151: Lecture 11, Pg 3 Lecture 11, ACT 2 When a car goes around a circular curve on a level road, a.no frictional force is needed because the car simply follows the road. b.the frictional force of the road on the car increases when the car’s speed decreases. c.the frictional force of the road on the car increases when the car’s speed increases. d.the frictional force of the road on the car increases when the car moves to the outside of the curve. e. there is no net frictional force because the road and the car exert equal and opposite forces on each other.

Physics 151: Lecture 11, Pg 4 Nonuniform Circular Motion Earlier we saw that for an object moving in a circle with nonuniform speed then a = a r + a t. arar atat What are F r and F t ?

Physics 151: Lecture 11, Pg 5 My match box car is going to do a loop the loop. What must be its minimum speed at the top so that it can make the loop successfully ?? Example Exercise 1 See Example 6.7

Physics 151: Lecture 11, Pg 6 My match box car is going to do a loop the loop. What must be its minimum speed at the bottom so that it can make the loop successfully ?? This is a difficult problem with just forces. We will deal with it with energy considerations in Ch. 9. Example Exercise 2 See Example 6.8

Physics 151: Lecture 11, Pg 7 My match box car is going to do a loop the loop. If the speed at the bottom is v B, what is the normal force at that point ?? Example Exercise 3 See Example 6.8

Physics 151: Lecture 11, Pg 8 I am building a roller coaster. I design it so that when I do a loop by myself I am weightless at the top, and I have speed v 1. Next my two friends get in with me so that the total weight of car and people doubles. How fast must the car go so we are still weightless at the top ?? Lecture 11, ACT 3 Nonuniform Circular Motion See Example 6.8 A) 1/2 v 1 B) v 1 C) 2 v 1 D) 4 v 1

Physics 151: Lecture 11, Pg 9 Lecture 10, ACT 4 l When a pilot executes a loop- the-loop (as in figure on the right) the aircraft moves in a vertical circle of radius R=2.70 km at a constant speed of v=225 m/s. Is the force exerted by the seat on the pilot: A) Larger B) Same C) Smaller then pilot’s weight (mg) at : I) the bottom and II) at the top of the loop. 2.7 km

Physics 151: Lecture 11, Pg 10 Lecture 10, ACT 4 Solution 2.7 km FcFc FcFc I) II) mg NINI N II FcFc l force exerted by the seat on the pilot is: A) Larger B) Same C) Smaller then pilot’s weight ANSWER (C) ANSWER (A)

Physics 151: Lecture 11, Pg 11 Example Gravity, Normal Forces etc. Consider a women on a swing: 1.When is the tension on the rope largest ? 2. Is it :A) greater than B) the same as C) less than the force due to gravity acting on the woman (neglect the weight of the swing) Animation

Physics 151: Lecture 11, Pg 12 Lecture 11, ACT 5 A rock attached to a string swings in a vertical circle. Which free body diagram could correctly describe the force(s) on the rock when the string is in one possible horizontal position ?

Physics 151: Lecture 11, Pg 13 Accelerated Reference Frames: The Accelerometer l Your first job is with General Motors. You are working on a project to design an accelerometer. The inner workings of this gadget consist of a weight of mass m that is hung inside a box that is attached to the ceiling of a car. You design the device with a very light string so that you can mathematically ignore it. The idea is that the angle the string makes with the vertical is determined by the car’s acceleration. Your preliminary task is to think about calibration of the accelerometer. First you calibrate the measurement for when the car travels on a flat road. See example 6-9: Train Car See text: 6-3

Physics 151: Lecture 11, Pg 14 Accelerated Reference Frames: The Accelerometer a  i See example 6-9: Train Car See text: We need to solve for the angle the plum bob makes with respect to vertical. First we will solve by using Newton’s Second Law and checking x and y components. Then we will consider other possible solution methods.

Physics 151: Lecture 11, Pg 15 Accelerometer Solving, i i: F X = T X = T sin  = ma j j: F Y = T Y  mg = T cos  mg = 0 l Eliminate T T gmggmg m amaama ji  TXTX TYTY  See text: 6-3 See example 6-9: Train Car T sin  = ma T cos  = mg 2. Draw a free-body diagram for the mass:

Physics 151: Lecture 11, Pg 16 Accelerometer – Other Thoughts 1 l Alternative solution using vectors : F l Find the total vector force F TOT : T gmggmg F F TOT  See text: 6-3 m T T (string tension) g mg (gravitational force)  See example 6-9: Train Car

Physics 151: Lecture 11, Pg 17 l Alternative solution using vectors F l Find the total vector force F TOT : F l Recall that F TOT = ma: l So amaama T gmggmg  See text: 6-3 m T T (string tension) g mg (gravitational force)  See example 6-9: Train Car Accelerometer – Other Thoughts 1

Physics 151: Lecture 11, Pg 18 Accelerometer – Other Thoughts 2 l Think of this problem from the point of view of the person inside the car. l This person sees the plumb bob making the same angle with respect to the ground, but detects no acceleration. a  See text: 6-3 See example 6-9: Train Car

Physics 151: Lecture 11, Pg 19 Accelerometer... l There must be some other force to balance the x component i i: F X = T X + F? = T sin  + F? = 0 j j: F Y = T Y  mg = T cos  mg = 0 l And we must put F? = -ma to get the same answer as before. l F? is known as a fictitious force.  See text: 6-3 See example 6-9: Train Car a T mgmg

Physics 151: Lecture 11, Pg 20 Lecture 11, ACT 6 Accelerated Reference Frames You are a passenger in a car and not wearing your seatbelt. Without increasing or decreasing speed, the car makes a sharp left turn, and you find yourself colliding with the right-hand door. Which is a correct description of the situation ? A) Before and after the collision there is a rightward force pushing you into the door. B) Starting at the time of the collision, the door exerts a leftward force on you. C) Both of the above. D) Neither of the above.

Physics 151: Lecture 11, Pg 21 Recap of today’s lecture l Forces and circular motion Ch l Accelerated reference frames Ch. 6.3 Reading for Mon., Ch pp. Work l Homework #4 (due 10/2/06 5 PM) : Problems from Ch.6 / on the webassign