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Let's break down the problem to find the coefficient of kinetic friction \(\mu_k\).

### Step 1: Analyze the energy conversion
- Initially, the block has potential energy stored in the compressed spring.
  The spring's potential energy is:
  \[
  E_{\text{spring}} = \frac{1}{2} k x^2
  \]
  where \(k\) is the spring constant and \(x\) is the compression distance.

- As the block slides across the table, it loses this energy due to work done against friction. The work done by friction is given by:
  \[
  W_{\text{friction}} = F_{\text{friction}} \cdot d = \mu_k m g \cdot d
  \]
  where \(d\) is the distance the block slides, \(\mu_k\) is the coefficient of kinetic friction, \(m\) is the mass of the block, and \(g\) is the acceleration due to gravity.

### Step 2: Energy conservation principle
The initial energy stored in the spring is completely dissipated by the work done by friction. Therefore, we equate the two expressions:
\[
\frac{1}{2} k x^2 = \mu_k m g \cdot d
\]

### Step 3: Solve for \(\mu_k\)
Rearranging the above equation to solve for the coefficient of kinetic friction:
\[
\mu_k = \frac{k x^2}{2 m g d}
\]

This is the expression for the coefficient of kinetic friction in terms of the given variables: \(k\) (spring constant), \(x\) (compression distance), \(m\) (mass), \(g\) (acceleration due to gravity), and \(d\) (sliding distance).

Would you like further clarification or details?

Here are five related questions you could explore:
1. How would the coefficient of kinetic friction change if the spring constant were doubled?
2. What would happen to the sliding distance \(d\) if the mass \(m\) were increased?
3. How does the work-energy principle apply in this problem?
4. Can you derive a similar expression if the surface had both kinetic and static friction?
5. How would the energy dissipation change if the spring were not perfectly elastic?

**Tip:** Remember that energy conservation is a powerful tool for analyzing problems with springs and friction, as mechanical energy can convert into other forms such as heat due to friction.

You push a block of mass m against a horizontal spring, compressing the spring a distance x. Then you release the block, and the spring sends it sliding across a tabletop. It stops a distance d from where you released it. Let k be the spring constant. What is the coefficient of kinetic friction between the block and the table?

Learn how to calculate the coefficient of kinetic friction between a block and a table using energy conservation principles with a compressed spring. This problem involves spring potential energy, work done by friction, and the work-energy principle. Suitable for students in Grades 11-12.

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