Agenda Today: Homework quiz, moment of inertia and torque Thursday: Statics problems revisited, rolling motion Reading: Start Chapter 8 in the reading Have to cancel office hours today: will have extra on Thursday morning (9-11:30) Chapter 6, Problem 18 A 200 g block on a 50-cm-long string swings in a circle on a horizontal, frictionless table at 75 rpm. a. What is the (linear) speed of the block? b. What is the tension in the string? Rotational Inertia Rotational inertia depends on Total mass of the object Distribution of the mass relative to axis Farther the mass is from the axis of rotation, the larger the rotational inertia. Rotational inertia ~ (mass) x (axis_distance)2 Rotational Inertia Depends upon the axis around which it rotates Easier to rotate pencil around an axis passing through it. Harder to rotate it around vertical axis passing through center. Hardest to rotate it around vertical axis passing through the end. Calculating Moment of Inertia Point-objects: I = Σm i r i 2 Solid sphere (through center): I = 2/5 MR 2 Hollow sphere (through center): I = 2/3 MR 2 Solid disk (through center): I = 1/2 MR 2 Hoop (through center) : I = MR 2 See textbook for more examples (pg. 216) Example: Hoop vs. Disk Imagine rolling a hoop and a disk of equal mass down a ramp. Which one would win? Which one is easier to rotate (i.e., has less rotational inertia)? 1
Torque Torque is the rotational analog of force. Same sign convention as other rotational quantities Depends on: Magnitude of Force (F) Direction of force Lever arm (r) Examples of Lever Arm Lever arm is amount of perpendicular distance to where the force acts: points from pivot point to location of force τ = r x F ( cross product ) Units: Nm Torque Forces are not always perpendicular to the lever arm! Torque definition picks out the perpendicular component of the force. τ = r perp F or = rf perp Torque is a vector quantity, can be treated in the same way as forces. A plumber attempts to tighted a bolt by exerting a force on the end of a wrench, perpendicular to the handle. Which has the largest torque? A. Wrench A: 20 cm, 10 N B. Wrench B: 40 cm, 5 N C. Wrench C: 20 cm, 5 N D. Wrench D: 40 cm, 10 N A plumber attempts to tighted a bolt by exerting a force on the end of a wrench, perpendicular to the handle. Which has the smallest torque? A. Wrench A: 20 cm, 10 N B. Wrench B: 40 cm, 5 N C. Wrench C: 20 cm, 5 N D. Wrench D: 40 cm, 10 N Right Hand Rule for Torque Point the fingers in the direction of the position vector Curl the fingers toward the force vector The thumb points in the direction of the torque 2
Torque Example A plumber attempts to loosen a bolt by pushing downward on a wrench angled 30º below the horizontal. If she can exert 75 N of force, compare the torque with the wrench at 30º to the torque if the wrench was completely horizontal. Revisiting Newton s Laws 1: Need a linear force to change an object s linear motion Need a torque to change an object s rotational motion Equilibrium: Linear: ΣF = 0 Rotational: Στ = 0 2: Translational acceleration ~ force, and ~ 1/mass Angular acceleration ~ torque, and ~ 1/rotational inertia Example: See-Saw Balancing 4 m? m Agenda Today: Center of gravity, torque and rotation worksheet, rolling without slipping Tuesday: Homework #7 quiz, statics and springs Center of Gravity Average position of all the mass in an object is called the center of mass (CG) of object. F g acts at the center of gravity important for torque! Calculating CG x cm = x 1 m 1 + x 2 m 2 + /(m tot ) y cm = y 1 m 1 + y 2 m 2 + /(m tot ) Center of mass/center of gravity can also be found experimentally by balancing/hanging an object. 3
Example: Where is the CG for each of these objects (assume rods are massless and 50 cm long, m= 1kg)? Example of a Free Body Diagram (Forearm) Example of a Free Body Diagram (Ladder) Example: Rank net torque about each pivot from least to greatest. The free body diagram shows the normal force and the force of static friction acting on the ladder at the ground The last diagram shows the lever arms for the forces Example Three trucks are parked on a slope. Which truck(s) tip over? A. Left truck B. Middle truck C. Right truck D. Both B and C E. All three Newton s Second Law for a Rotating Object Σ τ = Iα The angular acceleration is directly proportional to the net torque The angular acceleration is inversely proportional to the moment of inertia of the object Combine with F = ma for full statics problems 4
Example: Rod with two masses Rolling without slipping Two points masses, m 1 = 1kg and m 2 = 2kg, are suspended from a massless rod of total length L = 1m. The rod s pivot point is fixed a distance d = 0.25m from the left end (see diagram on the board). Find a group of people (no more than 3) to work with, and get as far through the problem as you can over the next 15 minutes. Discuss your approach with your group, and follow the general problemsolving strategy. Raise your hand if your group gets stuck, and I will come around to help. Rolling without slipping If the object completes one rotation, its center will move a linear distance of exactly one circumference: Δx = 2πr This gives us a relationship between linear velocity (of the center of the object) and angular velocity: v = 2πr/Δt = ωr Rolling without Slipping Applications: string unwinding from a cylinder or pulley Wheels on the road: connect linear motion and rotation Careful: don t apply v= rω blindly, or for any old location on the rolling object! Ask yourself if it makes sense for the problem. Example: Spinning your tires Consider a spot on the top of a car tire with radius 65 cm. If the car is stuck in the mud and the wheels just spin, what is happening? What is different when the tires get traction and start to roll without slipping? - Location of pivot point - Speed of the point at the top of the tire 5