Question

There are four equations :

#1 Tcos60 - Ncos30 = m(a - cos60)

#2 mg - Tsin60 - Nsin30 = m(a sin60)

#3 T + Ncos30 + Tcos60 - Tsin30 = Ma

#4 N1 = Nsin30 + Tsin60 + Tcos30

Find : T , a , N , N1

Given : m = 2 kg , M = 8kg

Answer 1

Tcos60 - Ncos30 = m(a - cos60) -------1

mg - Tsin60 - Nsin30 = m(a sin60)------2

T + Ncos30 + Tcos60 - Tsin30 = Ma ------3

N1 = Nsin30 + Tsin60 + Tcos30-----4

Re-writing the equations after plugging the values.

T/2-√3N/2 + 1 = 2a --------1

40 - √3T/2 - N/2 =√3a --------2

T + √3N/2 = 8a --------3

N1 = N/2 + √3T/2 --------4

__________________________

Adding equation (1) and (3)..
we get :-
a = 3T+2/20 ----(i) to

Now put value of a in equation 1&3

-5T/2 -√3N/2 = 19 -------1'
-11T +√3N/2 = 80 -------2'
adding both

we get
-27/2 = 99
=> T = -22/3

now put the value of T in equation 1&3

we get
-22/3 + √3/2N = 8a -------2"
-11/3 -√3/2N = 2a-1 -------1"

adding both the equations

we get a = -1 .

Put the value of a and T in equation (1)

we get :-

-11/3 -√3N/2 = -8
-11/3+8 = √3N/2
13/3 = √3N/2
=> N = 26/3√3 .

Put the value of N and T in equation (4) .

N1 = 13/3✓3 - 11√3/3 -11√3/3
=>
N1 = 13/3√3 - 22√3/3

=> N1 = 53/3√3.