Question

1. two blocks, with masses ( m_1 = 8 , \text{kg} ) and ( m_2 = 12 , \text{kg} ), are connected by a string and move across a rough surface (( mu_k = 0.4 )). the tension in the string between the masses is ( t = 44 , \text{n} ).

a. find the acceleration of each object.
b. find the force applied to ( m_2 ).

Explanation:

Step1: Calculate friction on $m_1$

$f_{k1} = \mu_k m_1 g = 0.4 \times 8 \times 9.8 = 31.36\ \text{N}$

Step2: Solve acceleration via $m_1$

Apply $\sum F = m_1 a$:
$T - f_{k1} = m_1 a$
$a = \frac{T - f_{k1}}{m_1} = \frac{44 - 31.36}{8} = 1.58\ \text{m/s}^2$

Step3: Calculate friction on $m_2$

$f_{k2} = \mu_k m_2 g = 0.4 \times 12 \times 9.8 = 47.04\ \text{N}$

Step4: Solve applied force via $m_2$

Apply $\sum F = m_2 a$:
$F_{app} - T - f_{k2} = m_2 a$
$F_{app} = T + f_{k2} + m_2 a = 44 + 47.04 + 12 \times 1.58 = 110\ \text{N}$

Answer:

a. The acceleration of each object is $\boldsymbol{1.58\ \text{m/s}^2}$
b. The applied force on $m_2$ is $\boldsymbol{110\ \text{N}}$