The problem you're working on involves using Newton's laws to analyze the motion of a block sliding down a smooth, frictionless inclined plane.

Part a: Determine the acceleration of the block.

Given:

• Mass of the block $m = 7.0 \, \text{kg}$
• Incline angle $\theta = 22.0^\circ$
• The plane is frictionless, so no frictional forces are acting on the block.

Using Newton's second law, the only force causing the block to accelerate down the incline is the component of its weight along the incline.

The gravitational force acting on the block is $F_g = m \cdot g$, where $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity.

The component of the gravitational force acting along the plane is:

$F_{\parallel} = m \cdot g \cdot \sin(\theta)$

According to Newton's second law, the net force along the incline causes the block to accelerate:

$F_{\parallel} = m \cdot a$

Therefore, the acceleration of the block is:

$a = g \cdot \sin(\theta)$

Substitute the values:

$a = 9.8 \, \text{m/s}^2 \cdot \sin(22.0^\circ)$

Let's calculate this.

Part b: Determine the block's speed when it reaches the bottom of the incline.

Given:

• The block starts from rest, so $v_0 = 0 \, \text{m/s}$
• Distance along the incline $d = 12.0 \, \text{m}$
• We know the acceleration from part a, and we need to find the final velocity $v_f$.

Using the kinematic equation:

$v_f^2 = v_0^2 + 2ad$

Since $v_0 = 0$, the equation simplifies to:

$v_f = \sqrt{2ad}$

Once we compute $a$ from part a, we can use this equation to find the final speed.

Let me calculate both results.### Results:

• Part a: The acceleration of the block as it slides down the plane is approximately $3.67 \, \text{m/s}^2$.
• Part b: The speed of the block when it reaches the bottom of the incline is approximately $9.39 \, \text{m/s}$.

Would you like further details or clarification?

Here are five follow-up questions you can consider:

1. How would the acceleration change if the incline angle were steeper?
2. What would happen to the final speed if friction were introduced on the incline?
3. How does the mass of the block affect its acceleration on a frictionless incline?
4. What additional forces would act if the surface were not frictionless?
5. How would you calculate the time it takes for the block to reach the bottom?

Tip: On frictionless inclined planes, acceleration depends only on the angle, not on the mass of the object.