A moving 4.80 kg block collides with a horizontal spring whose spring constant is 243 N/m.

The block compresses the spring a maximum distance of 5.50 cm from its rest position. The coefficient of kinetic friction between the block and the horizontal surface is 0.390. What is the work done by the spring in bringing the block to rest?

How much mechanical energy is being dissipated by the force of friction while the block is being brought to rest by the spring?

What is the speed of the block when it hits the spring?

You know the final PE. You know Ff (mu, m, g) Work done be the spring is INT Ffdx.

The sum of these is initial KE, so you can find initial speed.

To find the work done by the spring in bringing the block to rest, we need to calculate the potential energy stored in the spring when it is compressed.

First, we need to find the force exerted by the spring. This can be done using Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement from its equilibrium position:

F = -kx

Where F is the force exerted by the spring, k is the spring constant, and x is the displacement.

Given:
Block mass (m) = 4.80 kg
Spring constant (k) = 243 N/m
Maximum compression (x) = 5.50 cm = 0.055 m

Calculating the force exerted by the spring:
F = -kx
F = -(243 N/m)(0.055 m)
F ≈ -13.365 N

Since the displacement is in the opposite direction of force, we use a negative sign.

Now, to calculate the work done by the spring, we use the formula:

Work (W) = 0.5kx^2

Where W is the work done by the spring.

W = 0.5kx^2
W = 0.5(243 N/m)(0.055 m)^2
W ≈ 0.367 J

Therefore, the work done by the spring in bringing the block to rest is approximately 0.367 J.

Moving on to the next question, let's calculate the mechanical energy dissipated by the force of friction. The mechanical energy is dissipated by friction can be calculated by finding the work done by friction.

The work done by friction can be found using the formula:

Work (W) = Force (F) × Displacement (d)

To calculate the force of friction, we need to determine the normal force acting on the block. The normal force (Fn) is equal to the weight of the block (mg), where g is the acceleration due to gravity (approximated as 9.8 m/s^2).

Given:
Block mass (m) = 4.80 kg
Coefficient of kinetic friction (μ) = 0.390
Displacement (d) = 0.055 m (maximum compression)

Calculating the normal force:
Fn = mg
Fn = (4.80 kg)(9.8 m/s^2)
Fn ≈ 47.04 N

Now, we can calculate the force of friction:
Ffriction = μFn
Ffriction = (0.390)(47.04 N)
Ffriction ≈ 18.35 N

Lastly, let's calculate the work done by friction:
W = Ffriction × d
W = (18.35 N)(0.055 m)
W ≈ 1.009 J

Therefore, the mechanical energy dissipated by the force of friction while the block is being brought to rest by the spring is approximately 1.009 J.

For the final question, we need to find the speed of the block when it hits the spring. To do this, we can use the principle of conservation of mechanical energy.

Assuming no other external forces act on the block-spring system (except during compression due to friction), the initial mechanical energy of the block is equal to the final mechanical energy of the block-spring system.

Initial mechanical energy (Kinetic energy) = Final mechanical energy (Potential energy of the spring)

The initial mechanical energy of the block is given by:

Initial mechanical energy = (1/2)mv^2

Where m is the mass of the block and v is the speed of the block.

Given:
Block mass (m) = 4.80 kg

Let's rearrange the equation to solve for v:

v = sqrt(2 * Initial mechanical energy / m)

Calculating the initial mechanical energy:
Initial mechanical energy = (1/2)m(v^2)
Initial mechanical energy = (1/2)(4.80 kg)(v^2)

Since the block is initially at rest, its initial mechanical energy is 0.

0 = (1/2)(4.80 kg)(v^2)

Simplifying the equation:
0 = 2.4 v^2

Dividing both sides by 2.4:
0 = v^2

Taking the square root of both sides:
v = 0

Therefore, the speed of the block when it hits the spring is 0.