Review Example 13 before attempting this problem A marble and acube are placed a

Question

Review Example 13 before attempting this problem
A marble and acube are placed at the top of a ramp. Starting fromrest at thesame height, the marble rolls and the cube slides (nokineticfriction) down the ramp. Determine the ratio of thecenter-of-massspeed of the cube to the center-of-mass speed of themarble at thebottom of the ramp. Review Example 13 before attempting this problem
A marble and acube are placed at the top of a ramp. Starting fromrest at thesame height, the marble rolls and the cube slides (nokineticfriction) down the ramp. Determine the ratio of thecenter-of-massspeed of the cube to the center-of-mass speed of themarble at thebottom of the ramp.

Explanation / Answer

Energy and Momentum are always conserved. Physicists hold these two conservation laws as mostsacred.
---- Conservation of Energy:
Ei = Ef Classically, in terms of mechanics, KEi + Ui + Emech(i) =KEf + Uf + Emech(f) Here, mechanical energy = 0. ---- And the initial kinetic energy of each object is zero (startfrom rest). Also, if we set the final potential = 0 (which we can alwayschose any such reference point), we are left with: Ui = KEf --------- The kinetic energy of a translating body is translationalkinetic energy. i.e., if I throw a rock in the air, it has translationalkinetic energy (it is translating), among other things (potentialenergy). Now, if I have a child's top on a table, and give it aspin (without translating it anywhere laterally), it will haverotational kinetic energy. ------ So a marble rolling down a ramp will have bothtranslational and rotational kinetic energy, while a sliding cube will have just translational kineticenergy. [In fact, in terms of classical mechanics, there are said tobe only two possible motions: translation and rotation.] ----- For cube: KEc = KEcubetrans =(1/2)Mv2 For marble: KEm = KEmarble trans + KEmarblerot = (1/2)Mv2 + (1/2)I2 I = moment of inertia = ang velocity = v/r -------- For a solid sphere, Isolid sphere = (2/5)Mr2 KEm = (1/2)Mv2 +(1/2)(2/5)Mr2(v2/r2) =(1/2)Mv2 + (1/5)Mv2 =(7/10)Mv2 and
KEc = (1/2)Mv2 Recalling the derived formula:
Ui = KEf (that is, all initial potential energy is finally converted tokinetic energy at some given final state) ---- For cube: Ui = KEf Mgy = (1/2)Mv2 vcube2 = 2gy vcube = (2gy)1/2 ---- ---- For marble: Ui = KEf Mgy = (7/10)Mv2 vmarble2 = (10/7)gy vmarble = [(10/7)gy]1/2 vmarble2 = (10/7)gy vmarble = [(10/7)gy]1/2 Ratio of CM speed of cube to CM speed of marble is: vcube/vmarble =(2gy)1/2/[(10/7)gy]1/2 =(14/10)1/2 = (7/5)1/2 = 1.18 That is, vcube = (1.18)vmarble = 18%faster than marble.

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