In Classical Dynamics of Particles and Systems (Thornton & Marion), Chapter 2 #4
ID: 1686431 • Letter: I
Question
In Classical Dynamics of Particles and Systems (Thornton & Marion), Chapter 2 #42, both the solutions manual and Cramster have R(theta) as part of the height. R(theta) is the arc length created by the angle theta in the drawing. How then is it also the hypotenuse of the tiny triangle, such that R(theta)sin(theta) is part of the height? Generally, please explain where R(theta) comes from in terms of the picture. Thanks, Jeff In Classical Dynamics of Particles and Systems (Thornton & Marion), Chapter 2 #42, both the solutions manual and Cramster have R(theta) as part of the height. R(theta) is the arc length created by the angle theta in the drawing. How then is it also the hypotenuse of the tiny triangle, such that R(theta)sin(theta) is part of the height? Generally, please explain where R(theta) comes from in terms of the picture. Thanks, JeffExplanation / Answer
We again use conservation of energy because there is no sliding of the rigid body (cylinder) over the inclined plane.No work is done by the kinetic friction if the body roll without slipping.We can also ignore the effects of rolling friction,provided that the cylinder and the surface on which they roll are perfectly rigid. The cylinder starts from rest at the top of an incline with height h,so K1 = 0,U1 = Mgh,and U2 = 0.The kinetic energy at the bottom of the incline is given by the equation K = (1/2)I1 * w^2 = (1/2)I_cm * w^2 + (1/2)MR^2w^2 = (1/2)I_cm * w^2 + (1/2)Mv_cm^2 When the cylinder roll without slipping,w = (v_cm/R).The moment of inertia of the cylinder is I = MR^2 From conservation of energy, K1 + U1 = K2 + U2 0 + Mgh = (1/2)Mv_cm^2 + (1/2)MR^2 * (v_cm/R)^2 = (1/2) * (1 + 1) * Mv_cm^2 so the speed at the bottom of the incline is v_cm = (2gh/1 + 1)^1/2 = (gh)^1/2 This is a fairly amazing result;the speed doesn't depend on either the mass M of the body or its radius R. How then is it also the hypotenuse of the tiny triangle, such that R(theta)sin(theta) is part of the height? Generally, please explain where R(theta) comes from in terms of the picture.