Answer:
The acceleration of
is ![a = 0.7156 m/s^2](https://tex.z-dn.net/?f=a%20%3D%20%200.7156%20m%2Fs%5E2)
Explanation:
From the question we are told that
The mass of first block is ![M_1 = 2.25 \ kg](https://tex.z-dn.net/?f=M_1%20%3D%20%202.25%20%5C%20kg)
The angle of inclination of first block is ![\theta _1 = 43.5^o](https://tex.z-dn.net/?f=%5Ctheta%20_1%20%3D%20%2043.5%5Eo)
The coefficient of kinetic friction of the first block is ![\mu_1 = 0.205](https://tex.z-dn.net/?f=%5Cmu_1%20%20%3D%200.205)
The mass of the second block is ![M_2 = 5.45 \ kg](https://tex.z-dn.net/?f=M_2%20%3D%205.45%20%5C%20kg)
The angle of inclination of the second block is ![\theta _2 = 32.5^o](https://tex.z-dn.net/?f=%5Ctheta%20_2%20%3D%20%2032.5%5Eo)
The coefficient of kinetic friction of the second block is ![\mu _2 = 0.105](https://tex.z-dn.net/?f=%5Cmu%20_2%20%3D%200.105)
The acceleration of
are same
The force acting on the mass
is mathematically represented as
![F_1 = T - M_1gsin \theta_1 - \mu_1 M_1 g cos\theta_1](https://tex.z-dn.net/?f=F_1%20%3D%20T%20-%20%20M_1gsin%20%5Ctheta_1%20-%20%5Cmu_1%20M_1%20g%20cos%5Ctheta_1)
=> ![M_1 a = T - M_1gsin \theta_1 - \mu_1 M_1 g cos\theta_1](https://tex.z-dn.net/?f=M_1%20a%20%3D%20T%20-%20%20M_1gsin%20%5Ctheta_1%20-%20%5Cmu_1%20M_1%20g%20cos%5Ctheta_1)
Where T is the tension on the rope
The force acting on the mass
is mathematically represented as
![F_2 = M_2gsin \theta_2 - T -\mu_2 M_2 g cos\theta_2](https://tex.z-dn.net/?f=F_2%20%3D%20%20M_2gsin%20%5Ctheta_2%20-%20T%20-%5Cmu_2%20M_2%20g%20cos%5Ctheta_2)
![M_2 a = M_2gsin \theta_2 - T -\mu_2 M_2 g cos\theta_2](https://tex.z-dn.net/?f=M_2%20a%20%3D%20%20M_2gsin%20%5Ctheta_2%20-%20T%20-%5Cmu_2%20M_2%20g%20cos%5Ctheta_2)
At equilibrium
![F_1 = F_2](https://tex.z-dn.net/?f=F_1%20%3D%20%20F_2)
So
![T - M_1gsin \theta_1 - \mu_1 M_1 g cos\theta_1 =M_2gsin \theta_2 - T -\mu_2 M_2 g cos\theta_2](https://tex.z-dn.net/?f=T%20-%20%20M_1gsin%20%5Ctheta_1%20-%20%5Cmu_1%20M_1%20g%20cos%5Ctheta_1%20%3DM_2gsin%20%5Ctheta_2%20-%20T%20-%5Cmu_2%20M_2%20g%20cos%5Ctheta_2)
making a the subject of the formula
![a = \frac{M_2 g sin \theta_2 - M_1 g sin \theta_1 - \mu_1 M_1g cos \theta - \mu_2 M_2 g cos \theta_2 }{M_1 +M_2}](https://tex.z-dn.net/?f=a%20%3D%20%20%5Cfrac%7BM_2%20g%20sin%20%5Ctheta_2%20-%20M_1%20g%20sin%20%5Ctheta_1%20-%20%5Cmu_1%20M_1g%20cos%20%5Ctheta%20-%20%5Cmu_2%20M_2%20g%20cos%20%5Ctheta_2%20%7D%7BM_1%20%2BM_2%7D)
substituting values ![a = \frac{(5.45) (9.8) sin (32.5) - (2.25) (9.8) sin (43.5) - (0.205)*(2.25) *9.8cos (43.5) - (0.105)*(5.45) *(9.8) cos(32.5) }{2.25 +5.45}](https://tex.z-dn.net/?f=a%20%3D%20%20%5Cfrac%7B%285.45%29%20%289.8%29%20sin%20%2832.5%29%20-%20%282.25%29%20%289.8%29%20sin%20%2843.5%29%20-%20%280.205%29%2A%282.25%29%20%2A9.8cos%20%2843.5%29%20-%20%280.105%29%2A%285.45%29%20%2A%289.8%29%20cos%2832.5%29%20%7D%7B2.25%20%2B5.45%7D)
=> ![a = 0.7156 m/s^2](https://tex.z-dn.net/?f=a%20%3D%20%200.7156%20m%2Fs%5E2)