Physics 111 Lecture 3 (Walker: 10.6, 11.1) Conservation of Energy in Rotation Torque March 30, 009 Lecture 3 1/4 Kinetic Energy of Rolling Object Total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies: The second equation makes it clear that the kinetic energy of a rolling object is a multiple of the kinetic energy of translation. Lecture 3 /4
Example: Rolling Disk 1.0 kg disk with radius of 10.0 cm rolls without slipping. The linear speed of disk is v = 1.41 m/s. (a) Find the translational kinetic energy. (b) Find the rotational kinetic energy. (c) Find the total kinetic energy. Kt = mv = (1.0 kg)(1.41 m/s) = 1.19 J 1 1 K = Iω = ( mr )( v/ r) = (1.0 kg)(1.41 m/s) = 0.595 J r 1 1 1 1 4 K = K + t K = r (1.19 J) + Σ (0.595 J) = 1.79 J Lecture 3 3/4 Question A solid sphere and a hollow sphere of the same mass and radius roll forward without slipping at the same speed. How do their kinetic energies compare? (a) K solid > K hollow (b) K solid = K hollow (c) K solid < K hollow (d) Not enough information to tell Lecture 3 4/4
Rolling Down an Incline Ki + Ui = K f + U f U i = mgh 1 K = mv (1 + I / mr ) f 1 mgh= mv (1 + I / mr ) v= gh + I mr /(1 / ) Lecture 3 5/4 Hollow Cylinder : ; I = mr v= gh Solid Cylinder: I = mr ; v = gh 1 4 3 Hollow Sphere: I = mr ; v= gh 6 3 5 Solid Sphere: I = mr ; v = gh 10 5 7 Lecture 3 6/4
Question Which of these two objects, of the same mass and radius, if released simultaneously, will reach the bottom first? Or, is it a tie? Assume rolling without slipping. (a) Hoop; (b) Disk; (c) Tie; (d) Need to know mass and radius. Lecture 3 7/4 The Great Downhill Race A sphere, a cylinder, and a hoop, all of mass M and radius R, are released from rest and roll down a ramp of height h and slope θ. They are joined by a particle of mass M that slides down the ramp without friction. Who wins the race? Who is the big loser? Lecture 3 8/4
The Winners Lecture 3 9/4 Example: Spinning Wheel Block of mass m attached to string wrapped around circumference of a wheel of radius R and moment of inertia I, initially rotating with angular velocity ω. The block rises with speed v = rω. The wheel rotates freely about its axis and the string does not slip. To what height h does the block rise? E = E E = mgh E = mv + I = mv + I( v/ R) = mv (1 + I / mr ) i i f f 1 1 1 1 1 ω v I 1 h = + g mr Lecture 3 10/4
Example: Flywheel-Powered Car Your vehicle s braking mechanism transforms translational kinetic energy into rotational kinetic energy of a massive flywheel. Flywheel has I = 11.1 kg m and a maximum angular speed of 30,000 rpm. At the minimum highway speed of 40 mi/h, air drag and rolling friction dissipate energy at 10.0 kw. If you run out of gas 15 miles from home, with flywheel spinning at maximum speed, can you make it back? = ( 30, 000 rev/min )( π rad/rev ) / (60 s/min)=3,14 rad/s ω K = Iω = (11.1 kg m )(3,14 rad/s) = 54.9 MJ 1 1 t = x/ v= (15 mi) / (40 mi/h) = 0.375 h = 1350 s Energy needed =W nc =(10 kw)(1350 s)=13.5 MJ You make it. Lecture 3 11/4 Torque (τ) To make an object start rotating, a force is needed; the position and direction of the force matter as well. The perpendicular distance r from the axis of rotation to the line along which the force acts is called the lever arm. Lecture 3 1/4
Torque From experience, we know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis. This is why we have long-handled wrenches, and why doorknobs are not next to hinges. Lecture 3 13/4 Torque We define a quantity called torque which is a measure of twisting effort. For force F applied perpendicular distance r from the rotation axis: The torque increases as the force increases and also as the perpendicular distance r increases. Lecture 3 14/4
Other Torque Units Common US torque unit: foot-pound (ft-lb) Seen in wheel lug nut torques for vehicles: Ford Aspire 1994-97 Ford Contour 1999-00 Ford Crown Victoria 000-05 85 ft-lbs 95 ft-lbs 100 ft-lbs Torque Amplifier Lecture 3 15/4 A longer lever arm is very helpful in rotating objects. Lecture 3 16/4
Finding the Lever Arm r Here, the lever arm for F A is the distance from the knob to the hinge; the lever arm for F D is zero; and the lever arm for F C is as shown. Lecture 3 17/4 Torque in Terms of Tangential Force The torque is defined as: Could also write this as τ = r F where r is the distance from the axis to where the force is applied and F is the tangential force component. Lecture 3 18/4
Torque This leads to a different definition of torque (as given in the textbook): where F is the force vector, r is the vector from the axis to the point where force is applied, and θ is the angle between r and F. (Angle measured CCW from r direction to F direction.) Lecture 3 19/4 Sign of Torque If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative. CCW: τ > 0 CW: τ < 0 Lecture 3 0/4
Two Interpretations of Torque Torque can be considered as the r force F acting at a perpendicular distance d = r sin φ (called the moment arm) from the pivot point r to the line of action through F. Thus, τ = F ( r sin φ) = Fd. Lecture 3 1/4 Two Interpretations of Torque Torque can be considered as the distance r from the pivot to the point r of action of the force F times the tangential force component Ft = Fsin φ. Thus, τ = rf ( sin φ) = rf. t Lecture 3 /4
Question Five different forces are applied to the same rod, which pivots around the black dot. Which force produces the smallest torque about the pivot? Lecture 3 3/4 End of Lecture 3 For Wednesday, read Walker, 11.-4 Homework Assignment 10b is due at 11:00 PM on Wednesday, April 1. Lecture 3 4/4