The problem in the image involves a block of mass $m = 5.30 \, \text{kg}$ that slides down a frictionless track, with various heights indicated at points $A$, $B$, and $C$. Here’s how we can approach the problem:

Given Data:

• Mass of block, $m = 5.30 \, \text{kg}$
• Height at point $A$, $h_A = 5.40 \, \text{m}$
• Height at point $B$, $h_B = 3.20 \, \text{m}$
• Height at point $C$, $h_C = 2.00 \, \text{m}$

Questions:

1. (a) Determine the block’s speed at points $B$ and $C$.
2. (b) Determine the net work done by the gravitational force as the block moves from point $A$ to point $C$.

Solution Approach:

Part (a): Speed at Points $B$ and $C$

Since the track is frictionless, we can use the conservation of mechanical energy. The total mechanical energy at each point remains the same, assuming no external forces other than gravity are acting.

1. Calculate Initial Potential Energy at Point $A$: $\text{Potential Energy at } A = m \cdot g \cdot h_A$ where $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity.

2. At Point $B$:

   • The potential energy at $B$ is $m \cdot g \cdot h_B$.
   • Using energy conservation: $\text{Kinetic Energy at } B = \text{Potential Energy at } A - \text{Potential Energy at } B$ $\frac{1}{2} m v_B^2 = m g h_A - m g h_B$
   • Solving for $v_B$: $v_B = \sqrt{2 g (h_A - h_B)}$

3. At Point $C$:

   • The potential energy at $C$ is $m \cdot g \cdot h_C$.
   • Using energy conservation: $\text{Kinetic Energy at } C = \text{Potential Energy at } A - \text{Potential Energy at } C$ $\frac{1}{2} m v_C^2 = m g h_A - m g h_C$
   • Solving for $v_C$: $v_C = \sqrt{2 g (h_A - h_C)}$

Part (b): Net Work Done by Gravitational Force from $A$ to $C$

The work done by the gravitational force is equal to the change in gravitational potential energy as the block moves from point $A$ to point $C$:

$\text{Work} = m g (h_A - h_C)$

Substitution of Values:

Using the values for $m$, $g$, $h_A$, $h_B$, and $h_C$, we can calculate $v_B$, $v_C$, and the net work done.

Would you like me to complete the calculations with specific numerical values, or do you have any questions about the approach?

Here are five related questions to further explore this topic:

1. How would the presence of friction affect the speed at points $B$ and $C$?
2. How does the conservation of energy principle apply in other types of motion?
3. What would happen to the work done by gravity if the block’s mass was doubled?
4. How would the speeds change if the track had a different height at point $A$?
5. How is the concept of work-energy theorem applied in this scenario?

Tip: Always confirm if energy is conserved in the system (e.g., no friction) before applying conservation of energy.