Let’s solve this problem step by step:
Step 1: Determine the initial potential energy stored in the spring.
The potential energy stored in a compressed or stretched spring is given by the formula:
[tex]\[ PE_{\text{spring}} = \frac{1}{2} k x^2 \][/tex]
where:
- [tex]\( k \)[/tex] is the spring constant (2000 N/m)
- [tex]\( x \)[/tex] is the compression distance (0.20 m)
So,
[tex]\[ PE_{\text{spring}} = \frac{1}{2} \times 2000 \, \text{N/m} \times (0.20 \, \text{m})^2 \][/tex]
[tex]\[ PE_{\text{spring}} = 0.5 \times 2000 \times 0.04 \][/tex]
[tex]\[ PE_{\text{spring}} = 40 \, \text{J} \][/tex]
Thus, the initial potential energy in the spring is 40.00000000000001 J.
Step 2: Calculate the friction force acting on the block.
The friction force can be calculated using the formula:
[tex]\[ F_{\text{friction}} = \mu \times m \times g \][/tex]
where:
- [tex]\( \mu \)[/tex] is the coefficient of kinetic friction (0.4)
- [tex]\( m \)[/tex] is the mass of the block (3.00 kg)
- [tex]\( g \)[/tex] is the acceleration due to gravity (9.81 m/s[tex]\(^2\)[/tex])
So,
[tex]\[ F_{\text{friction}} = 0.4 \times 3.00 \, \text{kg} \times 9.81 \, \text{m/s}^2 \][/tex]
[tex]\[ F_{\text{friction}} = 0.4 \times 3.00 \times 9.81 \][/tex]
[tex]\[ F_{\text{friction}} = 0.4 \times 29.43 \][/tex]
[tex]\[ F_{\text{friction}} = 11.772 \, \text{N} \][/tex]
The friction force acting on the block is 11.772000000000002 N.
Step 3: Use the work-energy principle to find the distance the block slides.
According to the work-energy principle, the work done by the friction force is equal to the initial potential energy stored in the spring:
[tex]\[ \text{Work}_{\text{friction}} = F_{\text{friction}} \times \text{distance} \][/tex]
[tex]\[ PE_{\text{spring}} = F_{\text{friction}} \times \text{distance} \][/tex]
So,
[tex]\[ \text{distance} = \frac{PE_{\text{spring}}}{F_{\text{friction}}} \][/tex]
[tex]\[ \text{distance} = \frac{40.00000000000001 \, \text{J}}{11.772000000000002 \, \text{N}} \][/tex]
[tex]\[ \text{distance} \approx 3.40 \, \text{m} \][/tex]
Thus, the block slides a distance of approximately 3.397893306150187 meters before coming to rest.
