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Practice Problem PHY 1114: Physics I • The diameter of your tires is 0.60 m. You make a 60 mile trip at a speed of 45 mph. Lecture 11: (a) During this trip, what was your tires’ angular Rotational MotioN speed? Recall 1 mph = 0.447 m/s (b) How many times did they revolve? Fall 2012 Kenny L. Tapp Think about it... Rigid Bodies, Translations, & Rotations A ball on the end of a string swings in a horizontal circle once every second. State whether the magnitude of each of the following quantities is zero, constant (but not zero), or changing: a. Velocity d. Angular Acceleration b. Angular Velocity e. Tangential Acceleration A rigid body, in which distances between particles are fixed, may have translational motion, rotational motion, or a combination. It may roll with or without slipping. Every point on a rotating rigid body has the same angular velocity ω! c. Centripetal Acceleration Rotational Forces Rotational Forces Consider pushing open a heavy door. Four forces are shown, all of equal strength. Which of these will be most effective at opening the door? The ability of a force to cause a rotation is dependent upon 3 factors: 1. The magnitude F of the force F1 Hinge F3 Far Side Cartoon by Gary Larsen F4 F2 2. The distance r from the axis which an object can rotate, to the point where the force F is applied 3. The angle at which the force F is applied Torque, Equilibrium, & Stability It takes a force to start an object rotating; that force is more effective the farther it is from the axis of rotation, and the closer it is to being perpendicular to the line to that axis. Quick Question 1 • In trying to open a stuck classroom door, Prof. Tapp pushes it at a point 0.8 m from the hinges with a 220 N force that is directed 20° away from being perpendicular to the door. What torque does Tapp exert on the door? Torque, Equilibrium, & Stability The perpendicular distance from the line of force to the axis of rotation is called the lever arm. The product of the force and the lever arm is called the torque. Quick Question 2 • The bolts on a car wheel require tightening to a torque specification of 90 mN. If a 0.20 m long wrench is used, what is the magnitude of the force required: (a) when the force is perpendicular to the length of the wrench? (b) when the force is at θ = 35° to the length of the wrench? Torque, Equilibrium, & Stability In order for an object to be in equilibrium, the net force on it must be zero, and the net torque on it must be zero as well. Rotational Dynamics The net torque on an object causes its angular acceleration. For a point particle, the relationship between the torque, the force, and the angular acceleration is relatively simple. Recall at = rα and at = F/m.....we can rewrite and combine into one equation: α = F/(mr) Recall 𝜏 = rF, thus F = 𝜏/r.....with this relationship, we can rewrite our equation for α: α = 𝜏/(mr2) We now have an equation that gives a relationship between torque on a single particle and its angular acceleration. Rotational Dynamics Moments of Inertia We can consider an extended object to be a lot of near-point objects (single particles) stuck together. Then the net torque is: Net torque is the cause of angular acceleration. The quantity inside the parentheses is called the object’s moment of inertia, I. Moment of inertia is the rotational equivalent of mass. Net torque is the cause of angular acceleration. Net torque is cause of angu acceleration Quick Question 3 Moments of Inertia • The engine in a small airplane is specified to have a torque of 500 Nm. This engine drives a 2.0 m-long, 40 kg single-blade propeller. On start-up, how long does it take the propeller to reach 2000 rpm? ebay.com This is a rod rotating about its center Solve for α Solve for I Solve for time knowing ∆ω Net torque is the cause of angular acceleration. Quick Question 4 Quick Question 4 • A 5.0 m tall telephone pole with a mass of 260 kg had just been hit by a tornado, mostly severing the pole at its base. The pole falls, rotating about the part still connected to the base. Estimate the pole’s angular acceleration when it has fallen 25° from the vertical. A falling telephone pole undergoes an angular acceleration due to a gravitational torque. The pole’s weight w is acting at the center of gravity. The torque on the pole is thus due only to the gravity and is rotating the pole in a counterclockwise direction, so the torque is positive. Note: Fg = w⊥= w sinθ = mg sinθ w⊥ L/2 θ=25° 𝜏 = r F = (L/2) Fg = (L/2) w sinθ w Note: Model the pole as a uniform rod rotating about one end. Tornado damage in Norman, OK from April 13, 2012 𝜏 = (mgL/2) sinθ α = 𝜏/I Quick Question 5 • A standing athlete lifts a barbell by moving on his forearms, which pivot at the elbow. The weight lifted in the curl is over 200 pounds (about 900 N). The distance from the tendon to the elbow joint is 4.0 cm and from the barbell to the elbow is 35 cm. Quick Question 5 For the forearm to be in static equilibrium, the net force and net torque on it must both be zero. Ft Lifting muscle (biceps) ∑Fy = Ft - Fe - Fb = 0 Tendon 4 cm 35 cm a. What is the tension in the tendon connecting the biceps muscle to the bone while a 900 N barbell is held stationary in this position? b. What is the force exerted by the elbow on the forearm bones? Recall 𝜏 = r F... 𝜏net = [Ft rt] - [Fe re] - [Fb rb] = 0 re = 0 cm Fe Fb rt = 4 cm rb = 35 cm