Question

Find the acceleration of the sliding block??

Answer 1

Explanation:

mg cos @=F

mg cos @=ma

gcos @ =a

Answer 2

Answer:

Explanation:

method 1

summation fx = 0

for this you need acceleration with respect to ground reference

let acceleration of wedge be A rightwards

and acceleration be a down the lncline

acceleration of wedge = A

acceleration of block = acceleration of wedge+acceleration of block with            respect to wedge

acceleration of block with respect to wedge = -acostheta i -asintheta j

therefore acceleration of block = A-acostheta i -asintheta j

summation fx=0

M(A)+m(A-acostheta) = 0

macostheta/m+M = A

now apply pseudo or equation along incline plane

method1 for further equation

mgsintheta+mAcostheta=a

method 2 for obtaining further equation

for this your axis should be along and perpendicular to incline

acceleration of block wrt wedge along and perpendicular to incline axis = -a

acceleration of wedge wrt to along and perpendicular to incline axis = Acostheta i -Asintheta j

acceleration of block =Acostheta-a i -Asintheta j

mgsintheta = m(a-Acostheta)

substituting the first result in second of one of the methods

mgsintheta = m(a-macos^2 theta / m+M)

mgsintheta = m(ma+Ma-macos^2 theta )/m+M

gsintheta = Ma+masin^2 theta / M+m

(M+m)gsintheta = Ma+masin^2 theta

(M+m)gsintheta / M+msin^2 theta = a

above acceleration is wrt to wedge

macostheta/m+M = A

gsinthetacostheta / M+msin^2 theta = A

for acceleration wrt to ground = A-acostheta i -asintheta j

put above values in it

method 2

inertial method

ground axis

let acceleration of wedge = A

and acceleration of block with respect to wedge = a

acceleration of block wrt to ground axis = A-acostheta i -a sintheta j

applying equations of block wrt to ground axis

mg-ncostheta = masintheta

nsintheta=m(acostheta-A)

apply equation for incline

nsintheta=MA

MA=m(acostheta-A)

A=macostheta/m+M

Substitute A in above equations

mg-ncostheta = masintheta

mg-MAcottheta = masintheta

mgsintheta-MAcostheta = masin^2 theta

mgsintheta-Mmacos^2 theta / M+m = masin^2 theta

gsintheta-Macos^2 theta / m+M = asin^2 theta

(M+m)gsintheta-Macos^2 theta =( m+M)asin^2 theta

(M+m)gsintheta = Ma+masin^2 theta

(M+m)gsintheta = a(M+msin^2 theta )

(M+m)gsintheta / M+msin^2theta = a

A=mgsinthetacostheta /M+msin^2 theta

for acceleration of block with respect to ground

A-acostheta i -asintheta j

substitute A and a in above

method 3 non inertial method along incline

apply equation of wedge

MA=Nsintheta

apply equation of block along incline by apply pseudo

mgsintheta+macostheta=mA

N+mAsintheta = mgcostheta

MA/sintheta+mAsintheta = mgcostheta

MA+mAsin^2 theta =mgsinthetacostheta

A(M+msin^2theta) = mgsinthetacostheta

A=mgsinthetacostheta / M+msin^2theta

method 4  non inertial along ground

acceleration of block wrt to wedge=a

acceleration of wedge = A

equation of incline

Nsintheta=MA

equation of block along incline and wrt to ground

for this apply pseudo

mgsintheta+mAcostheta =ma

N+masintheta=mgcostheta

MA/sintheta +masintheta = mgcostheta

MA+masin^2 theta = mgsinthetacostheta

substitute constraint relation you will get same

method 5 inertial wrt to incline

acceleration of block wrt to block be a

acceleration of wedge be A

acceleration block wrt to ground wrt to incline axis

=Acostheta-a

acceleration of wedge wrt to incline = Acostheta i -Asintheta j

mgsintheta = m(a-Acostheta)

mgcostheta-N =mAsintheta

mgcostheta - MA/sintheta = mAsintheta

mgsinthetacostheta-MA = mAsin^2 theta

mgsinthetacostheta/M+msin^2 theta = A

by constraint help

a=(m+M)sintheta / M+msin^2 theta