PHY183-Lecture27pre - Rolling without Slipping Lets assume...

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1 March 13, 2012 Physics for Scientists&Engineers 1 1 Physics for Scientists & Engineers 1 Spring Semester 2012 Lecture 27 Rolling Objects and Torque March 13, 2012 Physics for Scientists&Engineers 1 2 Rolling without Slipping Let’s assume that we have a round object with radius R rolling without slipping This object has special relationships between its linear and angular quantities Displacement and angular displacement Velocity and angular velocity Acceleration and angular acceleration r = R θ v = R ω a = R α March 13, 2012 Physics for Scientists&Engineers 1 3 Rolling without Slipping (2) If an object rolls without slipping, the linear velocity of the point of contact with the surface is zero. Instantaneous axis of rotation Instantaneous velocity of any point on the object: is the position of a point on the object relative to the instantaneous axis of rotation vr r  March 13, 2012 Physics for Scientists&Engineers 1 4 Rolling without Slipping (2) . Take a look at two points directly above the axis of rotation Object is rolling to the right and the angular velocity is clockwise (into the screen) V cm =R ω V top =2R ω =2V cm March 13, 2012 Physics for Scientists&Engineers 1 5 Rolling and Kinetic Energy A rolling object has both translational kinetic energy and rotational kinetic energy We use our knowledge of the relationship between the linear and angular quantities to get K = K trans + K rot = 1 2 mv 2 + 1 2 I 2 K = 1 2 mv 2 + 1 2 I 2 = 1 2 mv 2 + 1 2 ( cR 2 m )( v / R ) 2 = 1 2 mv 2 + 1 2 mv 2 c = (1 + c ) 1 2 mv 2 (0 < c 1) March 13, 2012 Physics for Scientists&Engineers 1 6 The Rolling Race The acceleration of an object by gravity is independent of the mass of the object (free fall) What about rolling? Does the mass matter? Does the radius matter? Let’s look at the case of three objects with the same mass, same radius, but different distribution of mass, rolling down an inclined plane A solid sphere A solid cylinder A hollow cylinder Which one will win?
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2 March 13, 2012 Physics for Scientists&Engineers 1 7 Explanation for Race We can write the energy for each object c sphere = 0.4, smallest denominator, highest v E = K + U = K 0 + U 0 K = U 0 U (1 + c ) 1 2 mv 2 = mg ( h 0 h ) v = 2 g ( h 0 h ) 1 + c c sphere = 2 5 c cylinder = 1 2 c tube 1 Sphere Rolling Down an Inclined Plane PROBLEM A solid sphere with mass 5.15 kg and radius 0.340 m starts from rest at a height 2.10 m above the base of an inclined plane and rolls down under the influence of gravity
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This note was uploaded on 04/02/2012 for the course PHY 183 taught by Professor Wolf during the Spring '08 term at Michigan State University.

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PHY183-Lecture27pre - Rolling without Slipping Lets assume...

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