Respuesta :
Answer:
A ) SOLID SPHERE
Explanation:
Moment of inertia of solid sphere = 2/5 M R²
= M K² , K is called radius of gyration
K = √2/5 R
Moment of inertia of solid cylinder = 1/2 M R²
= M K² , K is called radius of gyration
K = 1 /√2 R
Moment of inertia of solid sphere = M R²
= M K² , K is called radius of gyration
K = R
For rolling on inclined plane , acceleration
a = [tex]\frac{gsin\theta}{1+\frac{K^2}{R^2} }[/tex]
Putting the value of K for solid sphere
a for solid sphere
a = g sinθ / ( 1 +2/5)
a = .714 g sinθ
Putting the value of K for solid cylinder
a for solid cylinder
a = g sinθ / ( 1 +1/2)
a = .666 g sinθ
Putting the value of K for hollow pipe
a for hollow pipe
a = g sinθ / ( 1 +1 )
a = . 5 g sinθ
So we see that acceleration a for solid sphere is greatest and a for hollow pipe is the least. Hence solid sphere will reach the bottom earliest and hollow pipe will reach the bottom the latest.
The correct option is option (A)
The solid sphere will reach the bottom first from the top of the inclined plane.
Inclined plane and moment of inertia:
The moment of inertia of the given objects is:
(i) Moment of inertia of a solid sphere
I = 2/5 M R²
I = MK²,
K is called the radius of gyration
K = √2/5 R
(ii) Moment of inertia of the solid cylinder
I = 1/2 M R²
I = MK²
K is called the radius of gyration
K = 1/√2 R
(iii) Moment of inertia of a hollow pipe:
I = MR²
I = MK²
K is called the radius of gyration
K = R
For rolling on the inclined plane without slipping, the acceleration is given by:
a = gsinθ/(1+K²/R²)
For solid sphere:
a = g sinθ / ( 1 +2/5)
a = 0.714gsinθ
For solid cylinder:
a = g sinθ / ( 1 +1/2)
a = 0.666gsinθ
For hollow pipe:
a = g sinθ / ( 1 +1 )
a = 0.5gsinθ
The acceleration for the solid sphere is the largest therefore, the solid sphere will reach the bottom first.
Learn more about moment of inertia:
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