The image depicts a physics problem involving two masses connected by a pulley system. Here's what the setup involves:

1. A mass $3m$ is placed on a horizontal surface.
2. A second mass $m$ is hanging vertically.
3. There is a coefficient of kinetic friction $\mu = 0.25$ between the $3m$ mass and the surface.
4. We are tasked with calculating:
   • The acceleration of the system.
   • The tension in the rope.

Let’s analyze the forces and solve:

1. Define forces on each mass:

For the $3m$ block (horizontal):

• Tension $T$ pulls the block to the right.
• Frictional force $f_k = \mu \cdot F_N = \mu \cdot 3m g$, where $F_N = 3m g$ is the normal force.
• Net force: $F_{\text{net},3m} = T - f_k = T - 0.25 \cdot 3m g$.

Newton's second law gives: $T - 0.25 \cdot 3m g = 3m a$

For the $m$ block (vertical):

• Gravitational force $F_g = m g$ pulls it downward.
• Tension $T$ opposes the downward pull.
• Net force: $F_{\text{net},m} = m g - T$.

Newton's second law gives: $m g - T = m a$

2. Set up equations and solve:

We now have two equations:

1. $T - 0.25 \cdot 3m g = 3m a$
2. $m g - T = m a$

Add these equations to eliminate $T$: $(T - 0.25 \cdot 3m g) + (m g - T) = 3m a + m a$

Simplify: $m g - 0.25 \cdot 3m g = 4m a$

$m g (1 - 0.75) = 4m a$

$0.25 m g = 4m a$

$a = \frac{0.25 g}{4} = \frac{g}{16}$

Using $g = 9.8 \, \text{m/s}^2$: $a = \frac{9.8}{16} \approx 0.6125 \, \text{m/s}^2$

3. Find the tension $T$:

Substitute $a = \frac{g}{16}$ into the second equation $m g - T = m a$: $m g - T = m \cdot \frac{g}{16}$

$T = m g - m \cdot \frac{g}{16}$

$T = m g \left(1 - \frac{1}{16}\right)$

$T = m g \cdot \frac{15}{16}$

Using $g = 9.8 \, \text{m/s}^2$: $T = m \cdot 9.8 \cdot \frac{15}{16} \approx 9.19 m \, \text{N}$

Final Results (with significant figures):

1. Acceleration of the system: $a \approx 0.613 \, \text{m/s}^2$ (3 significant figures).
2. Tension in the rope: $T \approx 9.19 m \, \text{N}$ (3 significant figures).

Would you like a step-by-step diagram explanation or clarification?

Related Questions:

1. What happens to the acceleration if the coefficient of friction increases?
2. How does the tension vary if $m$ and $3m$ are interchanged?
3. How would the system behave if the pulley had mass and friction?
4. How does the angle of the surface affect the calculations if it's inclined?
5. What is the work done by tension over a displacement of 5 meters?

Tip: Always double-check units and ensure the number of significant figures matches the input data!