Question

1. two blocks, with masses ( m_1 = 2 ) kg and ( m_2 = 8 ) kg, are connected by a string and move across a rough surface (( mu_k = 0.3 )). a force ( f = 50 ) n is applied to block ( m_2 ).

a. find the acceleration of each object.

b. find the tension force in the string between two objects.

Explanation:

Step1: Calculate total friction force

First, find the normal force for each block (equal to weight, $N = mg$), then total friction $f_k = \mu_k(m_1 + m_2)g$.
$f_k = 0.3 \times (2 + 8) \times 9.8 = 0.3 \times 10 \times 9.8 = 29.4\ \text{N}$

Step2: Find net applied force

Subtract total friction from applied force.
$F_{\text{net}} = F_{\text{app}} - f_k = 50 - 29.4 = 20.6\ \text{N}$

Step3: Calculate system acceleration

Use Newton's second law $a = \frac{F_{\text{net}}}{m_1 + m_2}$.
$a = \frac{20.6}{2 + 8} = \frac{20.6}{10} = 2.06\ \text{m/s}^2$

Step4: Find tension on block $m_1$

Tension provides net force for $m_1$ (overcoming its friction and accelerating it): $T = m_1a + \mu_k m_1 g$.
$T = 2 \times 2.06 + 0.3 \times 2 \times 9.8 = 4.12 + 5.88 = 10\ \text{N}$

Answer:

a. The acceleration of each object is $\boldsymbol{2.06\ \text{m/s}^2}$
b. The tension force in the string is $\boldsymbol{10\ \text{N}}$