Physics 111. Lecture 21 (Walker: ) Rolling Rotational Inertia Oct. 21, Rolling Motion. Rolling Motion

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Physics 111 Lecture 1 (Walker: 10.4-6) Rolling Rotational Inertia Oct.

1 Physics 111 Lecture 1 (Walker: ) Rolling Rotational Inertia Oct. 1, 009 Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds: Lecture 1 1/8 Lecture 1 /8 Rolling Motion We can consider rolling motion to be a combination of pure rotational and pure translational motion: Rolling Motion We may also consider rolling motion at any given instant to be a pure rotation at rate ω about the point of contact of the rolling object. Lecture 1 3/8 Lecture 1 4/8

2 Example: A Rolling Tire Car with tires of radius 3 cm drives at speed of 55 mph. What is angular speed ω of the tires? ω = v (55 mph)(0.447 m/s/mph) 77 rad/s r = (0.30 m) = Lecture 1 5/8 Rotational Kinetic Energy & Moment of Inertia (I) For this point mass m, Define the moment of inertia I of a point mass m located at distance r from rotation axis: I mr Lecture 1 6/8 Rotational Kinetic Energy & Moment of Inertia We can also write the rotational kinetic energy as R Where I, the moment of inertia, is given by Example: A Dumbbell Use definition of moment of inertia to calculate that of a dumbbell-shaped object with two point masses m separated by distance of r and rotating about a perpendicular axis through their center of symmetry. for a set of masses with mass m i located at distance r i from rotation axis. I, sometimes called rotational inertia, is the rotational equivalent of mass. Lecture 1 7/8 I = mr = mr + m r = mr i i 11 Lecture 1 8/8

3 Example: Grindstone Grindstone of radius r = m being used to sharpen an axe. If linear speed of stone is 1.50 m/s and stone s rotational kinetic energy is 13.0 J, what is its moment of inertia. Moment of Inertia of a Hoop ω = v/ r = (1.50 m/s) / (0.610 m) =.46 rad/s K (13.0 J) K = Iω I = = = 4.30 kg m ω (.46 rad/s) 1 Lecture 1 9/8 i i i ( i) I = mr = mr = m R = MR All the mass of hoop is at same distance R from center of rotation, so its moment of inertia is same as that of point mass rotated at the same distance. Lecture 1 10/8 Question The T-shaped object is rotated about each axis shown. For which axis does it have the largest moment of inertia? Moments of Inertia Lecture 1 11/8 Lecture 1 1/8

The Parallel-Axis Theorem* Kinetic Energy of Rolling Object Total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies: I = I + Md If you know the moment of

4 The Parallel-Axis Theorem* Kinetic Energy of Rolling Object Total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies: I = I + Md If you know the moment of inertia I cm of an object about its center of mass, you can get I about any other parallel axis by adding Md, where M is object s mass and d is cm The second equation makes it clear that the kinetic energy of a rolling object is a multiple of the kinetic energy of translation. separation of the axes. Lecture 1 13/8 Lecture 1 14/8 Example: Rolling Disk Question 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) = J r K K K (1.19 J) (0.595 J) 1.79 J 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 = + = + = Σ t r Lecture 1 15/8 Lecture 1 16/8

5 Rolling Down an Incline 0 0 K + U = K + U i i f f Conservation of mechanical energy. U i = mgh 1 K = mv (1 + I / mr ) f 1 mgh = mv (1 + I / mr ) v= gh + I mr /(1 / ) Lecture 1 17/8 Hollow Cylinder : ; I = mr v = gh Solid Cylinder: I = mr ; v= gh Hollow Sphere: I = mr ; v = gh Solid Sphere: I = mr ; v = gh Lecture 1 18/8 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 1 19/8 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 1 0/8

6 The Winners Lecture 1 1/8 Example: Spinning Wheel Block of mass m attached to string wrapped around circumference of 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? Ei = Ef E f = mgh E = mv + Iω = mv + I( v / R) = mv (1 + I / mr ) i v I h = 1+ g mr Lecture 1 /8 Example: Flywheel-Powered Car Your vehicle s braking mechanism transforms translational kinetic energy into rotational kinetic energy of massive flywheel. Flywheel has I = 11.1 kg m and maximum angular speed 30,000 rpm. At 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? ω = ( 30, 000 rev/min)( π rad/rev ) / (60 s/min)=3,14 rad/s K = Iω = (11.1 kg m )(3,14 rad/s) = 54.9 MJ 1 1 t = x/ v= (15 mi) / (40 mi/h) = h = 1350 s Energy needed =W nc =(10 kw)(1350 s)=13.5 MJ You make it. Lecture 1 3/8 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 1 4/8

Torque From experience, we know that a force will be more effective at rotating an object such as a nut or door if our hand is not too close to the axis.

7 Torque From experience, we know that a force will be more effective at rotating an object such as a nut or door if our hand is not too close to the axis. Torque (τ) We define a quantity called torque which is a measure of twisting effort. For force F applied perpendicular distance r from the rotation axis: This is why we have long-handled wrenches, and why doorknobs are not next to The torque increases as the force increases and also as the perpendicular distance r increases. Lecture 1 hinges. 5/8 Lecture 1 6/8 Other Torque Units Common US torque unit: foot-pound (ft-lb) Seen in wheel lug nut torques for vehicles: Ford Aspire ft-lbs Ford Contour ft-lbs Ford Crown Victoria ft-lbs End of Lecture 1 For Wednesday Oct. 8, read Walker Homework assignment PS10b due on WebAssign by 11:00 p.m. on Tues., Oct. 7. Torque Amplifier Lecture 1 7/8 Lecture 1 8/8

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013 PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be