A hollow sphere of radius 0.130 m, with rotational inertia I = 0.0514 kg·m 2 abo

Question

A hollow sphere of radius 0.130 m, with rotational inertia I = 0.0514 kg·m2 about a line through its center of mass, rolls without slipping up a surface inclined at 30.8° to the horizontal. At a certain initial position, the sphere's total kinetic energy is 33.0 J. (a) How much of this initial kinetic energy is rotational? (b) What is the speed of the center of mass of the sphere at the initial position? When the sphere has moved 0.540 m up the incline from its initial position, what are (c) its total kinetic energy and (d) the speed of its center of mass?

Explanation / Answer

let mass of the sphere be m kg.

moment of inertia of a hollow sphere is given by I=2*m*r^2/3 where r is radius of the sphere

then 2*m*0.13^2/3=0.0514

==>m=4.562 kg

part a:

let speed at this location is v m/s

then angular speed=linear speed/radius=v/0.13 rad/s

total kinetic energy=linear kinetic energy+rotational kinetic energy

==>33=0.5*mass*speed^2+0.5*rotational inertia*angular speed^2

==>33=0.5*4.562*v^2+0.5*0.0514*(v/0.13)^2

==>33=3.80171*v^2

==>v=2.9462 m/s

then rotational kinetic energy=0.5*0.0514*(v/0.13)^2=13.2 J

percentage of total kinetic energy=13.2/33=40%

part b:

speed of the center of mass of the sphere at initial position is 2.9462 m/s

part c:

as there are no friction forces, total energy is conserved.

when it rises 0.54 m along the incline, total height rise as compared to intiial position=0.54*sin(30.8)=0.2765 m

then rise in potential energy=mass*g*height=4.562*9.8*0.2765=12.3618 J

then decrease in kinetic eenrgy=-12.3618 J (in order to keep total energy constant)

hence kinetic energy at that location=33-12.3618=20.6382 J

part d:

let speed of center of mass is v.

then

total kinetic energy=linear kinetic energy+rotational kinetic energy

==>20.6382=0.5*mass*speed^2+0.5*rotational inertia*angular speed^2

==>20.6382=0.5*4.562*v^2+0.5*0.0514*(v/0.13)^2

==>20.6382=3.80171*v^2

==>v=2.33 m/s

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.