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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 Wheel has an angular acceleration of 3.0 rad/s2 and an initial angular speed of 2.00 rad/s. In a time of 2s it has rotated thorough an angle (in radian) of:

  • 6

  • 10

  • 12

  • 4


B.

10

Angular acceleration is time derivative of angular speed and angular speed is time derivative of angular displacement.

By definition,

straight alpha space equals space dω over dt
ie comma space dω space equals space αdt
So comma space if space in space time space straight t space the space angular space speed space of space straight a space body space changes space from space straight omega subscript straight o space to space straight omega
integral subscript straight omega subscript straight o end subscript superscript straight omega space dω space equals space integral subscript 0 superscript straight r space straight alpha space dt
If space straight alpha space is space constant
straight omega space minus straight omega subscript straight o space equals space αt
straight omega space equals space straight omega subscript straight o space plus αt
Now comma space as space by space definition
straight omega space equals space dθ over dt
Eq space left parenthesis straight i right parenthesis space becomes
dθ over dt space equals space space straight omega subscript straight o space plus αt
dθ space equals space left parenthesis straight omega subscript straight o space plus αt right parenthesis dt
So space straight m space if space in space time space straight t space angular space displacement space is space straight theta
integral subscript 0 superscript straight theta dθ space equals space integral subscript 0 superscript straight t space left parenthesis straight omega subscript straight o space plus αt right parenthesis space dt
Given comma space straight alpha space equals space 3.0 space rad divided by straight s squared comma space space equals space 2.0 space rad divided by straight s comma space straight t space equals space 2 straight s
Hence comma space straight theta space equals space 2 space straight x space 2 space plus space 1 half space straight x space 3 space left parenthesis 2 right parenthesis squared
or space straight theta space equals space 4 plus 6 space equals space 10 space rad

Angular acceleration is time derivative of angular speed and angular speed is time derivative of angular displacement.

By definition,

straight alpha space equals space dω over dt
ie comma space dω space equals space αdt
So comma space if space in space time space straight t space the space angular space speed space of space straight a space body space changes space from space straight omega subscript straight o space to space straight omega
integral subscript straight omega subscript straight o end subscript superscript straight omega space dω space equals space integral subscript 0 superscript straight r space straight alpha space dt
If space straight alpha space is space constant
straight omega space minus straight omega subscript straight o space equals space αt
straight omega space equals space straight omega subscript straight o space plus αt
Now comma space as space by space definition
straight omega space equals space dθ over dt
Eq space left parenthesis straight i right parenthesis space becomes
dθ over dt space equals space space straight omega subscript straight o space plus αt
dθ space equals space left parenthesis straight omega subscript straight o space plus αt right parenthesis dt
So space straight m space if space in space time space straight t space angular space displacement space is space straight theta
integral subscript 0 superscript straight theta dθ space equals space integral subscript 0 superscript straight t space left parenthesis straight omega subscript straight o space plus αt right parenthesis space dt
Given comma space straight alpha space equals space 3.0 space rad divided by straight s squared comma space space equals space 2.0 space rad divided by straight s comma space straight t space equals space 2 straight s
Hence comma space straight theta space equals space 2 space straight x space 2 space plus space 1 half space straight x space 3 space left parenthesis 2 right parenthesis squared
or space straight theta space equals space 4 plus 6 space equals space 10 space rad

1714 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

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

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

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

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