A steel cable lying flat on the floor drags a 20 kgblock across a horizontal, frictionless floor. A 100 Nforce applied to the cable causes the block to reach a speed of 4.2 m/s in a distance of 1.9 m. What is the mass of the cable?

To find the mass of the steel cable, we can approach this problem by applying Newton's second law and the work-energy principle.

### Step 1: Apply Work-Energy Theorem
The work done on the block by the force applied to the cable results in the kinetic energy of the system. The work-energy theorem states:

\[
W = \Delta K
\]

Where:
- \( W \) is the work done by the force.
- \( \Delta K \) is the change in kinetic energy.

The change in kinetic energy for the block and the cable system is given by:

\[
\Delta K = \frac{1}{2}(m_{\text{block}} + m_{\text{cable}})v^2
\]

Where:
- \( m_{\text{block}} = 20 \, \text{kg} \) (mass of the block),
- \( m_{\text{cable}} \) is the unknown mass of the cable,
- \( v = 4.2 \, \text{m/s} \) is the final speed of the block and cable system.

### Step 2: Calculate the Work Done by the Force
The work done on the system by the applied force is:

\[
W = F \cdot d
\]

Where:
- \( F = 100 \, \text{N} \) is the applied force,
- \( d = 1.9 \, \text{m} \) is the distance over which the force is applied.

Now, let's calculate the work done:

\[
W = 100 \, \text{N} \times 1.9 \, \text{m} = 190 \, \text{J}
\]

### Step 3: Use the Work-Energy Theorem
From the work-energy theorem:

\[
190 \, \text{J} = \frac{1}{2}(20 \, \text{kg} + m_{\text{cable}})(4.2 \, \text{m/s})^2
\]

Simplifying:

\[
190 = \frac{1}{2}(20 + m_{\text{cable}}) \times 17.64
\]

Multiply both sides by 2:

\[
380 = (20 + m_{\text{cable}}) \times 17.64
\]

Now divide both sides by 17.64:

\[
\frac{380}{17.64} = 20 + m_{\text{cable}}
\]

\[
m_{\text{cable}} = \frac{380}{17.64} - 20
\]

Let me calculate this.The mass of the steel cable is approximately \( 1.54 \, \text{kg} \).

Would you like more details on the solution or any further explanations?

Here are 5 related questions you might consider:
1. How does friction affect the acceleration in such a system if present?
2. What if the force was applied at an angle instead of horizontally? How would that change the outcome?
3. How does increasing the block's mass affect the speed it reaches after being dragged a certain distance?
4. What would happen if the cable itself had significant elasticity? 
5. How would you calculate the force if the block was on an inclined plane?

**Tip**: The work-energy theorem is a powerful tool to solve problems where forces cause changes in speed, especially in frictionless or near-frictionless conditions!

A steel cable lying flat on the floor drags a 20 kg block across a horizontal, frictionless floor. A 100 N force applied to the cable causes the block to reach a speed of 4.2 m/s in a distance of 1.9 m. What is the mass of the cable?

Discover how to calculate the mass of a steel cable dragging a 20 kg block on a frictionless surface with a 100 N applied force. This detailed solution involves applying the Work-Energy Theorem and Newton's second law, suitable for advanced high school students studying physics.

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