Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

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Mechanical Engineering : Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

The document Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Engineering Mechanics - Notes, Videos, MCQs & PPTs.
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1.- Kinematics. Types of motion:Translation, Rotation about a fixed axis , General Plane Motion, Motionabout a fixed pint, General Motion

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Exercise: Distinguish between curvilinear translation and rotation about a fixed axis

Translation.A motion is said to be a translation if any straight line inside the body keeps the same direction during the movement. All the particles forming the body move along parallel paths. If these paths are straight lines, the motion is said a rectilinear translation; if the paths are curved lines, the motion is a curvilinear motion

Rotation about a fixed axis. The particles forming the rigid body move in parallel planes along circles centered on the same fixed axis. If this axis, called the axis of rotation intersects the rigid body, the particles located on the axis have zero velocity and zero acceleration

 

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

General Plane Motion. Any plane motion which is neither a translation or a rotation is referred as a general plane motion. Plan motion is that in which all the particles of the body move in parallel planes. Translation and rotation are plane motions.

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Motion about a fixed point. The three-dimensional motion of a rigid body attached at a fixed point, for example, the motion of a top on a rough floor, is known as motion about a fixed point.

General Motion Any motion of a rigid body which does not fall in any of the cathegories above described. 

Exercise: Identify different types of motion

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Translation. Motion equations  Fig. 15.1 Fig. 15.7 pag 918

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Conclusion: A rigid body in translation can be considered as a particle

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

will be constant in magnitude (rigid body) and in direction (translation motion), then

the derivative of  Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev is zero

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

When a rigid body is in translation all the points of the body have the same velocity and the same acceleration.

Rotation about a fixed axis. 

Motion equations. Velocity 

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Angular velocity and angular acceleration are invariants. They are the same for all points of the solid. They are a characteristic of the rotating motion of the solid

Basic relationships curvilinear motion

arc angle x radius

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Where angle is in radians!!!

Exercise: A compact disk rotating at 500 rev/min is scanned by a laser that begins at the inner radius of about 2.4 cm and moves out the edge at 6.0 cm. Which is the linear (tangential) velocity of the disk where the laser beam strikes: (a) at the beginning of scanning and (b) at the end?. The same for acceleration  

Rotation about A Fixed Axis. Motion equations. Acceleration  Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Representative slab

Rotation about a Fixed Axis. The Vector Motion Equations

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Vector expressions for velocity and acceleration in rotation about a fixed axis

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

 

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

The red arrow shows the angular velocity of the horizontal gear 1. Draw the angular velocity for the other gear, 2 and 3. Solve the problem with quantitative values:

ω1 = 500 rev/min; R1 = 2 cm

ω2 = ? rev/min; R2 = 5 cm; R´2=10 cm

ω3 = ? rev/min; R4 = 10 cm; 

 

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev The bucket falls from the rest with a constant linear acceleration of 0.3 g. (a) Estimate the speed of the bucket after 5 seconds and the fallen distance. (b) Compute the angular acceleration of the pulley © How fast will it rotate after 5 s.

 

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev Gear 1 rotates clockwise at angular velocity of 12 rad/s. How fast will gear 2 and 3 rotate. Data: R1:5 cm; R2:10 cm; R3:20 cm.

 

General Plane Motion. Any general plane motion can be considered as a translation plus a rotation

Euler´s Theorem

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

General Plane Motion. Any general plane motion can be considered as a translation plus a rotation

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Angular velocity and angular acceleration of rod are independent of the selected point to rotate 

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRevMoment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Rolling without slipping.

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

As the wheel of radius R rotates through angle θ, the point of the contact between the wheel and the plane moves a distance s that is related with θ by s= θ R. 

If there is no sliding, the distance traveled by point C is exactly the same s.

Rolling without slipping.

s = θR

vc = ω R

ac = αR

  Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Rolling with slipping. An object slides and rolls

Rolling with slipping.

s ≠θ R

vc ≠ωR

ac≠ α R

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

Find the angular velocity of sliding stair of length 3 m, when the velocity of contact point with the soil is 3 m/s. The angles between the stair and the floor is 45º

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

A bicycle travels with a speed of 40 km/h. How fast the cycle rider pedals in rev/min?. Data: Sprocket radius: 2.5 cm; Front gear radius: 10 cm; rear wheel radius: 40 cm

Moment of Force on a Rigid Body Mechanical Engineering Notes | EduRev

The slider-crank mechanism converts the rotational motion of crank in linear motion of slider. Find the relationship between the angular velocity of crank and the linear velocity of slider piston 

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