Mechanics 2 Flashcards
What is the axis of rotation?
The centers of the circles traced by the rotating particles lie on a straight line called the axis of rotation which is fixed and perpendicular to the planes of the circles.
Define rigid body
A rigid body is one whose geometric shape and size remain unchanged under the action of any external force.
Define moment of a force or torque
The ability of a force to produce rotational motion is called moment of a force or torque.
What kind of motion does a couple produce? Why?
Rotational only, not translational.
Net force acting on the rigid body is zero.
What is rotational inertia of a body?
The quantity that measures the inertia of rotational motion of the body is called the rotational inertia or moment of inertia of the body.
Expression for kinetic energy of a rigid body rotating with constant angular speed
E= (1/2) Iω*ω
Define moment of inertia
The moment of inertia of a rigid body about an axis of rotation is defined as the sum of the product of the mass of each particle and the square of its perpendicular distance from the axis of rotation.
What factors influence moment of inertia of a body?
Mass, shape and size of the body
Distribution of mass of the body about the axis of rotation
Position and orientation of the axis of rotation
Give the physical significance of I
- F=ma and T=Ia
2. Et=(1/2)mvv and Er=(1/2)Iωω
Similarities and difference between mass and moment of inertia
Mass and MI remain constant.
But unlike mass, MI depends on position of axis of rotation, shape and size of the body, and distribution of mass about the axis of rotation.
How are flywheels and grinding wheels designed?
Flywheel: more mass distributed near the rim that at centre region near the axis. Also large diameter. So, MI increases.
Grinding wheel : large mass, moderate diameter. So, MI large, maintains motion and does not stop quickly when set into motion.
Define Radius of Gyration
If the whole mass of body ‘M’ is at a radial distance ‘K’ from the axis of rotation then the moment of inertia remains same about that axis of rotation, then ‘K’ is called radius of gyration of the body.
Radius if gyration - Equation
I = MKK
Physical significance of radius of gyration
Depends on shape and size of the body.
Measures distribution of mass about the axis of rotation.
Small value- mass distributed closer to axis of rotation and MI small
Large value- mass distributed at large distance from axis of rotation so MI large
Direction of torque and angular acceleration
Directed Parallel to the axis of rotation of the body
Equation for total KE for rolling motion
E= (1/2) Mvv[1+ (KK/(R*R))]
Equation for velocity of rolling motion down an inclined plane
V= root( 2gh/(1+(KK/(RR))))
Equation for acceleration of rolling motion down and inclined plane
a= gsin(θ) / (1+(KK/(RR)))
State the theorem of parallel axes
The theorem of perpendicular axes states that the moment of inertia of a body about any axis is equal to the sum of the it’s moment of inertia about a parallel axis passing through its centre of mass and the product of its mass and the square of the perpendicular distance between the two perpendicular axes.
State the theorem of perpendicular axes
The theorem of perpendicular axes states that the Moment of Inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of its moment of Inertia about two mutually perpendicular axes concurrent with the perpendicular axis and lying in the plane of the laminar body.
Limitation of theorem of perpendicular axes
Only for laminar bodies
MI about a thin uniform rod about an axis passing through its centre of mass and perpendicular to its length
I = MLL/12
MI of a thin ring about a transverse axis passing through its centre
I=MRR
Moment of inertia of a ring about its diameter
I=MRR/2