Sliding and rolling, Mechanical Engineering

Assignment Help:

Sliding and Rolling

Consider the following example. A cylinder of mass M and radius R is held at rest on a plane inclined at an angle to  817_download.png  to the horizontal. When the cylinder is released it moves down the slope. Let us discuss the motion of the cylinder.


When the cylinder is released, the component Mg sin  817_download.png  of its weight Mg along the incline causes the cylinder to slide down the slope. If the force of friction between cylinder and incline is ƒ, we get

Mg sin  817_download.png  - ƒ = Mac

where, ac is the linear acceleration of CM of the cylinder.

The frictional force provides a rotational torque about the CM axis of the cylinder:

ƒR = Ic  817_download.png 

where, Ic = Mk2 is the moment of inertia of the cylinder about its CM axis (which is a principal axis); k is radius of gyration. The axis of rotation does not change its direction in space, or with respect to cylinder.

with constant acceleration, ac, the CM velocity increases with time as,

vc = v0 + ac t = ac t                       (v0 = 0)

Simultaneously, with constant angular acceleration  817_download.png , the angular velocity of rotation of the cylinder increases as,

ω = ω0 +  817_download.png  t =  817_download.png  t                         (ω0 = 0)
    
Rolling: If a=  817_download.png  R, then vP = 0 independent of t. That is, if,

1984_download (1).png 

then the object performs pure rolling throughout the motion, there is no sliding.

However, the frictional force cannot exceed its limiting value, ƒmax =  928_download (2).png  N =  928_download (2).png  Mg cos  817_download.png . Hence, so long as ƒ given is less than (or at most equal to) ƒmax, the frictional force would acquire the value required to support rolling. Thus, the object will perform only pure rolling, if

1515_download (3).png 

For the cylinder, k2 = R2/2. Hence, we find that if  928_download (2).png  ≥ 1/3 tan  817_download.png  then a cylinder kept on an inclined plane will purely roll down. The acceleration of its CM then is,

451_download (4).png 

= 2/3 g sin  817_download.png        (for cylinder)

The rolling is therefore equivalent to translation of CM with acceleration ac and rotation about CM axis with angular acceleration  817_download.png = ac/R. The kinetic energy of rolling object (the cylinder) is, therefore

2386_download (5).png 

  2485_download (9).png 

where we used the fact that vc = ω R in rolling, and Ic = Mk2.

For rolling cylinder, we have  312_download (6).png 
    
Rolling with slipping: If  1299_download (7).png  then the value of ƒ is not physically possible. This in turn implies that ac =  817_download.png  R, orvc = ω R are not possible and therefore the object can never perform pure rolling. The object therefore will always slide and also rotate, as it moves down the slope.

Now, the frictional force is equal to force of sliding friction, i.e.

ƒ =  928_download (2).png  Mg cos

Both the linear and angular accelerations (and velocities) are given by the same relations as before:

ac =  (sin  817_download.png  -  817_download.png  cos  817_download.png ) g, vc = ac t

426_download (8).png 

The object translates with same linear acceleration as in the case of pure sliding, but also simultaneously rotates.


Related Discussions:- Sliding and rolling

Wet saturated steam - thermodynamics, Wet saturated steam -  thermodynamic...

Wet saturated steam -  thermodynamics: we t saturated steam Wet steam is a 2-phase mixture comprising of boiling water particles and dry steam in equilibrium state. Its for

State the planning the foundation for reciprocating engines, State the Plan...

State the Planning the foundation for reciprocating engines i)  The eccentricity of  the common centre of gravity of machine and foundation referred to the centroid of base are

Problem type-aspects of scheduling , Problem Type Aforementioned illust...

Problem Type Aforementioned illustration relates to the class of non-cyclic scheduling problems characterized via the subsequent features. (a) The Flexible Manufacturing Sys

What is the spindle and drill, What is the spindle and drill The spindl...

What is the spindle and drill The spindle and drill head assembly the taper hole can be create on radial drilling machine with some special attachment likewise quill(The spindl

Strength of materials, derive the torsion formula as (torsion divided by ra...

derive the torsion formula as (torsion divided by radius) is equal to (modulus of rigidity multiplied by twist of angle all divided by the length of a cylinder)

Determine the break power of the engine, If a Laboratory experiment, the fo...

If a Laboratory experiment, the following data were recorded with rope Break dynamometer : Diameter of fly-wheel = 1.2 m Diameter of the rope = 12.5 mm Speed = 200 r.p.m.

Entropy, What is entropy microscopically

What is entropy microscopically

Shielding gas-argon - hydrogen mixtures, Argon - Hydrogen Mixtures A...

Argon - Hydrogen Mixtures Argon - hydrogen mixtures are employed in special cases, such as mechanised welding of light gauge stainless steel tubing, in which the hydrogen do

Hydro turbine, in hydro turbine vaccum is present in casing or not? & if no...

in hydro turbine vaccum is present in casing or not? & if not then how suction is take place?

Discover modulus of elasticity of the composite, Discover modulus of elasti...

Discover modulus of elasticity of the composite: A composite of glass fibres and epoxy contain all fibres laid along the length and is needed to carry a stress of 1200 MPa. Th

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd