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System of Particles and Rotational Motion

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Physics Part I

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Physics

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CBSE Gujarat Board Haryana Board

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Class 10 Class 12

A disc and a sphere of the same radius but differnt masses roll off two inclined planes of the same altitude and length. which one of the two objects gets to the bottom of the plane first?

  • Sphere

  • Both reach at the same time

  • Depends on their masses

  • Disc


A.

Sphere

a) Acceleration of an object rolling down an inclined plane is given by,

straight a space equals space fraction numerator straight g space sin space straight theta over denominator 1 plus begin display style straight I over mr squared end style end fraction
where comma space straight theta space equals space angle space of space inclination space of
the space inclined space plane

m = mass of the object,
I = moment of Inertia about the axis through the centre of mass.

For space disc comma
straight I over mr squared space equals space fraction numerator bevelled 1 half space m r squared over denominator m r squared end fraction equals 1 half
For solid sphere,
straight I over mr squared equals fraction numerator bevelled 2 over 5 space m r squared over denominator m r squared end fraction space equals space 2 over 5
For hollow sphere,

straight I over mr squared equals fraction numerator bevelled 2 over 3 space m r squared over denominator m r squared end fraction equals 2 over 3

therefore space straight a subscript disc space equals space fraction numerator straight g space sin space straight theta over denominator 1 plus begin display style 1 half end style end fraction
space space space space space space space space space space space space space equals 2 over 3 space straight g space sinθ
space space space space space space space space space space space space space equals space 0.66 space straight g space sin space straight theta
straight a subscript solid space sphere end subscript space equals space fraction numerator straight g space sin space straight theta over denominator 1 plus begin display style 2 over 5 end style end fraction
space space space space space space space space space space space space space space space space space space space equals space 5 over 7 space straight g space sin space straight theta
straight a subscript hollow space sphere end subscript space equals space fraction numerator straight g space sin space straight theta over denominator 1 plus begin display style 2 over 3 end style end fraction
space space space space space space space space space space space space space space space space space space space space space equals space 3 over 5 space straight g space sin space straight theta
space space space space space space space space space space space space space space space space space space space space equals space 0.6 space straight g space sin space straight theta
Clearly comma space
straight a subscript solid space sphere end subscript space greater than space straight a subscript disk greater than straight a subscript hollow space sphere end subscript 

Therefore, the given sphere is a solid sphere.

asolid sphere = ahollow sphere > adisk

a) Acceleration of an object rolling down an inclined plane is given by,

straight a space equals space fraction numerator straight g space sin space straight theta over denominator 1 plus begin display style straight I over mr squared end style end fraction
where comma space straight theta space equals space angle space of space inclination space of
the space inclined space plane

m = mass of the object,
I = moment of Inertia about the axis through the centre of mass.

For space disc comma
straight I over mr squared space equals space fraction numerator bevelled 1 half space m r squared over denominator m r squared end fraction equals 1 half
For solid sphere,
straight I over mr squared equals fraction numerator bevelled 2 over 5 space m r squared over denominator m r squared end fraction space equals space 2 over 5
For hollow sphere,

straight I over mr squared equals fraction numerator bevelled 2 over 3 space m r squared over denominator m r squared end fraction equals 2 over 3

therefore space straight a subscript disc space equals space fraction numerator straight g space sin space straight theta over denominator 1 plus begin display style 1 half end style end fraction
space space space space space space space space space space space space space equals 2 over 3 space straight g space sinθ
space space space space space space space space space space space space space equals space 0.66 space straight g space sin space straight theta
straight a subscript solid space sphere end subscript space equals space fraction numerator straight g space sin space straight theta over denominator 1 plus begin display style 2 over 5 end style end fraction
space space space space space space space space space space space space space space space space space space space equals space 5 over 7 space straight g space sin space straight theta
straight a subscript hollow space sphere end subscript space equals space fraction numerator straight g space sin space straight theta over denominator 1 plus begin display style 2 over 3 end style end fraction
space space space space space space space space space space space space space space space space space space space space space equals space 3 over 5 space straight g space sin space straight theta
space space space space space space space space space space space space space space space space space space space space equals space 0.6 space straight g space sin space straight theta
Clearly comma space
straight a subscript solid space sphere end subscript space greater than space straight a subscript disk greater than straight a subscript hollow space sphere end subscript 

Therefore, the given sphere is a solid sphere.

asolid sphere = ahollow sphere > adisk

3089 Views

What is the significance of defining the center of mass of a system?

The motion of n particle system can be reduced to one particle motion.

An equivalent single point object would enable us to discuss the overall motion of the system. 
794 Views

Is it necessary that there should be matter at the centre of mass of system?

No, it is not necessary that there be matter at the centre of mass of the system.

For e.g., if two equal point masses are separated by certain distance, the centre of mass lies at the mid point of two point masses and there is no mass at that point.
730 Views

Define centre of mass.

Centre of mass of a body or a system of bodies is a point at which the entire mass of the body or system is supposed to be concentrated. 
1449 Views

What is the need of centre of mass?

Newton’s second law of motion is strictly applicable to point masses only. To apply the Newton's law of motion to rigid bodies, the concept of centre of mass is introduced.

The concept of centre of mass of a system enables us to discuss overall motion of the system by replacing the system by an equivalent single point object. 
911 Views

Is it necessary for centre of mass to lie within the body?

No, centre of mass needs not to lie within the body. It is not necessary that the total mass of the system be actually present at the centre.

The position of the centre of mass is calculated using the usual Newtonian type of equations of motion. 
943 Views