PreClass Notes: Chapter 5, Sections 5.1-5.3 From Essential University Physics 3 rd Edition by Richard Wolfson, Middlebury College 2016 by Pearson Education, Inc. Narration and extra little notes by Jason Harlow, University of Toronto This video is meant for University of Toronto students taking PHY131. 2012 Pearson Education, Inc. Slide 1-1 Outline Image from http://www.decodedscience.com/side-effect-of-rolling-an-airplane-aircraft-yaw/7209 Why does an airplane tip when it s turning? R.Wolfson 5.1 Problem Solving with Newton s Second Law 5.2 Objects Connected by Ropes and Pulleys 5.3 Circular Motion 2012 Pearson Education, Inc. Slide 1-2 1
Problem Solving Strategy 5.1 Interpret the problem to make sure Newton s second law is the relevant concept. Identify which objects are of interest and think about the forces on each object. Identify connections between objects and the constraints on their motion. Draw a free-body diagram for each object. Develop your solution plan by writing Newton s second law in components for each object. Execute your plan and solve the equations. Remember that the constraints are like equations. Assess your solution to see whether it makes sense. Think about units, special cases. 2012 Pearson Education, Inc. Slide 1-3 Example 5.1 A skier of mass m = 65 kg glides down a slope at an angle of θ = 32, as shown. Find (a) the skier s acceleration and (b) the force the snow exerts on the skier. The snow is so slippery that you can neglect friction. 2012 Pearson Education, Inc. Slide 1-4 2
Example 5.1 2012 Pearson Education, Inc. Slide 1-5 Example 5.1 2012 Pearson Education, Inc. Slide 1-6 3
Got it? A ball of mass m is suspended by a string from the ceiling inside an elevator. If the elevator is moving upward with a constant speed, the tension in the string A. is greater than mg. B. is equal to mg. C. is less than mg. D. depends on the speed of the elevator. 2012 Pearson Education, Inc. Slide 1-7 Tension Figure (a) shows a heavy safe hanging from a rope The combined pulling force of billions of stretched molecular springs is called tension Tension pulls equally in both directions Figure (b) is a very thin cross section through the rope This small piece is in equilibrium, so it must be pulled equally from both sides 2012 Pearson Education, Inc. Slide 1-8 4
Example 5.2 To protect her 17 kg pack from bears, a camper hangs it from ropes between two trees, as shown. What s the tension in each rope? 2012 Pearson Education, Inc. Slide 1-9 Example 5.2 2012 Pearson Education, Inc. Slide 1-10 5
The Massless String Approximation Often in problems the mass of the string or rope is much less than the masses of the objects that it connects. In such cases, we can adopt the following massless string approximation: 2012 Pearson Education, Inc. Slide 1-11 Got it? A rope is tied to a hook that is attached to a wall. If you pull the rope with a 1-N force, the force exerted by the hook on the rope A. is greater than 1 N. B. is less than 1 N. C. is equal to 1 N. D. cannot be determined from the information given. 2012 Pearson Education, Inc. Slide 1-12 6
Pulleys Block B drags block A across a frictionless table as it falls The string and the pulley are both massless There is no friction where the pulley turns on its axle Therefore, T A on S = T B on S 2012 Pearson Education, Inc. Slide 1-13 Acceleration Constraints If two objects A and B move together, their accelerations are constrained to be equal: a A = a B This equation is called an acceleration constraint Consider a car being towed by a truck In this case, the acceleration constraint is a Cx = a Tx = a x Because the accelerations of both objects are equal, we can drop the subscripts C and T and call both of them a x 2012 Pearson Education, Inc. Slide 1-14 7
Acceleration Constraints Sometimes the acceleration of A and B may have different signs Consider the blocks A and B in the figure The string constrains the two objects to accelerate together But, as A moves to the right in the +x direction, B moves down in the y direction In this case, the acceleration constraint is a Ax = a By 2012 Pearson Education, Inc. Slide 1-15 Example 5.4 A 73 kg climber finds himself dangling over the edge of an ice cliff, as shown. Fortunately, he s roped to a 940 kg rock located 51 m from the edge of the cliff. Unfortunately, the ice is frictionless, and the climber accelerates downward. What s his acceleration? 2012 Pearson Education, Inc. Slide 1-16 8
Example 5.4 2012 Pearson Education, Inc. Slide 1-17 Example 5.4 2012 Pearson Education, Inc. Slide 1-18 9
Dynamics of Uniform Circular Motion An object in uniform circular motion is not traveling at a constant velocity in a straight line. Consequently, the particle must have a net force acting on it Without such a force, the object would move off in a straight line tangent to the circle. The car would end up in the ditch! 2012 Pearson Education, Inc. Slide 1-19 Example 5.6 2012 Pearson Education, Inc. Slide 1-20 10
Example 5.6 2012 Pearson Education, Inc. Slide 1-21 Example 5.6 2012 Pearson Education, Inc. Slide 1-22 11
Example 5.6 2012 Pearson Education, Inc. Slide 1-23 Example 5.7 The Dragon Fire roller coaster at Canada s Wonderland features a double loop section. One of the loops is shown, and the radius of curvature at the top is 6.3 m. What s the required speed for a roller coaster at the top of the loop if the normal force from the track is to be zero (neither pushing nor pulling)? 2012 Pearson Education, Inc. Slide 1-24 12
Example 5.7 2012 Pearson Education, Inc. Slide 1-25 Example 5.7 2012 Pearson Education, Inc. Slide 1-26 13