A solid cylinder (disk) and a hollow cylinder are rolling with the same center-o

Question

A solid cylinder (disk) and a hollow cylinder are rolling with the same center-of-mass velocity v = 2 m/s on a level surface towards an incline. Both cylinders have the same radius R and mass M. The moments of inertia are:
Solid cylinder Is = MR2/2
Hollow cylinder Ih = MR2

(a) What is true about the kinetic energy when the cylinders are rolling on level ground?
(i) they have the same translational kinetic energy
(ii) they have the same rotational kinetic energy
(iii) they have the same total kinetic energy
[1] only (i) [2] only (ii) [3] (i) and (ii) [4] (i) and (iii) [5] (i), (ii), and (iii)

(b) Which conservation law will allow you to calculate the final height? Conservation of:
[1] momentum [2] mechanical energy [3] kinetic energy [4] angular momentum
[5] moment of inertia

(c) Calculate the maximum height the solid cyliner will reach before it rolls down again; please use g = 10 m/s2.
[1] 0.2 m [2] 0.3 m [3] 0.4 m/s [4] 0.6 m [5] none of these

(d) Calculate the maximum height the hollow cyliner will reach before it rolls down again.
[1] 0.2 m [2] 0.3 m [3] 0.4 m/s [4] 0.6 m [5] none of these

Explanation / Answer

Part(A) (i) translational kinetic energy is given by (1/2)mv2 . Given that they posses the same mass and velocity, their translational kinetic energies will be identical. (ii) Rotational kinetic energy is given (1/2)I2 given that they posses the same radius, their angular velocity will be the same. Consider each object moment of Inertia, for a solid cylinder: (1/2)mr2 and for a hollow cylinder: mr2. So for the solid sphere:(1/2)(1/2)mr22 =>(1/4)mr22 and for the hollow sphere:(1/2)mr22. So their rotational kinetic energies will be different.(iii) Total kinetic energy is translational kinetic energy plus rotational kinetic energy. Their rotational kinetic energies diifer and their translational kinetic energies are identical; thus, their totatl kinetic energy is different. So the answer is [1]

Part(B) The answer is the conservation of mechanical energy, because this concept contains both potential and kinetic energies. Gravitational potential energy will be needed to calculate the final height since gravitational potential energy=mgh.

Part(C) Eo+|Wnc|=Ef where Wnc=0J so Eo=Ef. Eo=(1/2)mv2+(1/2)I2 where I=(1/2)mr2. Ef=mgh

(1/2)mv2+(1/4)mr22=mgh; r=v; (1/2)mv2+(1/4)mv2=mgh =divide both sides by mass>

(1/2)v2+(1/4)v2=gh =collect like terms> (3/4)v2=gh =divide both sides by g> (3/4)(v2/g)=h

plug in g=10(m/s2) v=2(m/s): (3/4)((2)2/(10))=(3/4)(4/(10))=(3/10)=.3 or h=0.3m

Part(D) Eo+|Wnc|=Ef where Wnc=0J so Eo=Ef. Eo=(1/2)mv2+(1/2)I2 where I=mr2. Ef=mgh

(1/2)mv2+(1/2)mr22=mgh =divide both sides by mass and r=v> (1/2)v2+(1/2)v2=gh =collect like terms>

v2=gh =divide both sides by g> (v2/g)=h plug in plug in g=10(m/s2) v=2(m/s): ((2)2/10)=(4/10) or h=0.4m

so part(A) is: [1], part(B) is: mechanical energy, part(C): is [2], and part(D) is: [3]

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