Exploring the Meaning of Equations Exploring the meaning of the relevant ideas and equations introduced recently. This week we ll focus mostly on Newton s second and third laws: Kinematics describes the development of the motion once velocities & accelerations are specified, kinematics tells us how the system will move as a result of those velocities and accelerations. Newton s Laws describe the cause of the motion; where the velocities and accelerations come from. When calculating the net force we ll often use a tool called a Free Body Diagram, which is a pictorial way of doing a vector sum: This tells us how to sum different forces and what the total result will be. Newton s second law then relates that total result to the motion of the system. Today we ll use most of the forces you are currently familiar with. In particular, we will use the constraint forces (e.g., T and N) that do not have formulas for them you have to use information in the problem to solve for them. 1
Problem Solving Strategy Practicing strategic analysis by going through the 3 steps 1) Read and translate to physics, make drawing (FBD); 2) Conceptual analysis; 3) Connect 1 & 2 to choose a strategy. Verify your answer with your TA for credit before you move on to the next problem. (Spring 04 Exam 1 #4) (74% correct) Two blocks, of mass M 1 = 8 kg and M 2 = 3 kg are in contact with each other on a frictionless floor. A horizontal force F = 72 Newtons is applied to block M 1 as shown. F M 1 M 2 What is the force F 2,1 of the block of mass M 1 on the block of mass M 2? 1. Use FBD 1 to determine acceleration 2. Use FBD 2 and acceleration to determine F 12 F 12 = F (M 2 /(M 1 +M 2 )) 2
(Fall 2003 Exam 1 #10) A 211 student runs to catch a Frisbee. Just when the Frisbee passes over her head it is moving at a speed of 4 m/s, and this is when she starts to run accelerating at a rate of 1 m/s 2. The Frisbee is decelerating at a rate of 1.5 m/s 2. (You may neglect the vertical motion of the Frisbee). How far from her initial position does the student catch up with the Frisbee? 1. Write down equations of motion for Frisbee and student (x as a function of time) 2. Set positions of Frisbee and student equal to each other and solve for time 3. Substitute time into equation of motion for student to determine distance t = 2v frisbee /(a student a frisbee ) (note: a frisbee is negative) x = (1/2)a student t 2 x = 5.12 m (Spring 06 Exam 1 #24) (68% correct) A ball of mass M = 0.2 kg is suspended from a string. The ball travels in a horizontal circle of radius R = 0.5 m at a constant speed of v = 1.5 m/s. What is the magnitude of the net force, F net, on the mass? 1. Determine acceleration from speed and radius of uniform circular motion 2. Use acceleration and Newton s second law to determine F net. F net = M v 2 /R (note: F net is horizontal = vector sum of T + Mg = Tsinθ) 3
T PHYS 100 (from 221) Newton s Laws Week8 (Spring 05 #7) (97% correct) A mass of 5 kg is suspended from the ceiling of an elevator by an ideal massless rope. The rope will break if the tension T exceeds T max = 60 N. The elevator moves upward with the acceleration a. What is the maximum vertical acceleration the elevator can have without breaking the rope? 1. Write down Newton s second law for M 2. Substitute T = T max to determine a max a max = (T max Mg)/M a T max = 60N 5kg g (Spring 07 Exam 1 #3) (85% correct) A mass m = 8.1 kg is suspended by two wires hanging from a beam. The wires are set up so that they hang at equal angles, θ, from the beam. The magnitude of the tension in one wire, T 1, is measured to be 109 N. θ θ T 1 T 2 m = 8.1 kg What is the angle, θ, in degrees? 1. Draw FBD and write down Newton s second law (note: T 1 = T 2 from symmetry) 2. Solve for θ sinθ = mg/(2t) 4
Problem Practice Practicing strategic analysis by going through the 3 steps 1) Read and translate to physics, make drawing (FBD); 2) Conceptual analysis; 3) Connect 1 & 2 to choose a strategy; 4) Follow your strategy and solve the problem; 5) Reflect did the strategy work? Does the result make sense? (Spring 05 #8) (81% correct) A mass of 5 kg is suspended from the ceiling of an elevator by an ideal massless rope. The rope will break if the tension T exceeds T max = 60 N. Now the elevator is moving in horizontal direction with the acceleration a. What is the maximum horizontal acceleration the elevator can have without breaking the rope? 1. Write down Newton s second law for M in vertical and horizontal directions 2. Solve for angle θ max when T = T max in vertical equation 3. Substitute T = T max and θ = θ max in horizontal eqn to determine a max cosθ max = mg/t max a max = T max sinθ max /m 5
(Fall 03 Exam 1 #18) (65% correct) A car, which weighs 2000 N, travels over a bumpy road with a constant speed. Gravity acts. The road at point B is in the shape of a circle with a radius of 100 meters. With what speed must the car travel such that the force that it exerts on the road at point B is 600N? B 100m 1. Draw FBD for car at B and write down Newton s second law 2. Set acceleration at B = v 2 /R 3. Solve for v from vertical equation using N = 600N v 2 = R (g (N/M)) A (Fall 08 Exam 1 #10) (% correct) What is the acceleration of the hanging mass? m a = g (m/(m + 2M)) 6
(Fall 03 Exam 1 #24) (67% correct) A 1000 kg car is driving with constant velocity up a hill inclined at an angle of 5 with respect to the horizontal. What is F net, the magnitude of the sum of all of the forces on the car? F net = 0 (acceleration = zero!!) (Spring 06 Exam 1 #21) (56% correct) A box is at rest on a table. What is the Newton s Third Law reaction force corresponding to the weight of the box? The force the box exerts on the Earth (Universal gravitation: we haven t done this one yet). Note: it is not equal to the normal force since both the wieight and the normal force are forces exerted ON THE BOX. 7
(Spring 04 Exam 1 #17) (78% correct) A student twirls a tennis ball of mass m at the end of a string of length l in a vertical plane. v m l g What is the minimum speed, v, required to just keep the string taut when the ball is at the top of its travel? v 2 = Rg You are lying on a hill that slopes at 45. You throw a ball at a right angle to the hill s surface at 6m/s. When the ball lands on the hill below you, how far is it from you? Geometrical challenge: Draw the picture! 1. determine the initial angle with respect to horizontal 2. write down equations of motion (constant velocity in horizontal and constant acceleration = -g in the vertical) 3. solve for position of intersection with the y = x line (the hill) 8