A moving 4.5 kg block collides with a horizontal spring whose spring constant is 226 N/m (see figure). The block compresses the spring a maximum distance of 8.0 cm from its rest position. The coefficient of kinetic friction between the block and the horizontal surface is 0.47. What is the work done by the spring in bringing the block to rest?

To find the work done by the spring in bringing the block to rest, we need to consider the different forces acting on the block.

First, we need to calculate the force of friction opposing the motion of the block. The formula for the force of friction is:

\(F_f = \mu \cdot N\)

Where:
\(F_f\) is the force of friction,
\(\mu\) is the coefficient of kinetic friction,
\(N\) is the normal force.

The normal force (\(N\)) is the force exerted by the surface on the block, and it is equal to the weight of the block since there is no vertical acceleration. The formula for weight is:

\(W = m \cdot g\)

Where:
\(W\) is the weight of the block,
\(m\) is the mass of the block,
\(g\) is the acceleration due to gravity.

Next, we calculate the work done by the spring. The formula for work is:

\(W = \frac{1}{2} \cdot k \cdot x^2\)

Where:
\(W\) is the work done by the spring,
\(k\) is the spring constant,
\(x\) is the maximum displacement of the spring.

Finally, we can calculate the work done by the spring in bringing the block to rest:

\(Work_{spring} = - \mu \cdot N \cdot d\)

Where:
\(Work_{spring}\) is the work done by the spring,
\(\mu\) is the coefficient of kinetic friction,
\(N\) is the normal force,
\(d\) is the displacement of the block.

Substituting the given values:

\(m = 4.5 \, \text{kg}\),
\(k = 226 \, \text{N/m}\),
\(x = 8.0 \, \text{cm} = 0.08 \, \text{m}\),
\(\mu = 0.47\).

First, calculate the weight of the block:

\(W = m \cdot g = 4.5 \, \text{kg} \cdot 9.8 \, \text{m/s}^2\)

Then, calculate the force of friction:

\(F_f = \mu \cdot N\)

Next, calculate the work done by the spring:

\(W = \frac{1}{2} \cdot k \cdot x^2\)

Finally, calculate the work done by the spring in bringing the block to rest:

\(Work_{spring} = - \mu \cdot N \cdot d\)

By substituting the calculated values, you can find the work done by the spring in bringing the block to rest.