To determine the final displacement of the particle from its starting point after it strikes the spring, we need to analyze the motion of the particle and the interaction with the spring. We will break this down into several steps: calculating the frictional force acting on the particle, determining the work done by friction, and then finding how far the spring compresses due to the particle's kinetic energy.
Step 1: Calculate the Frictional Force
The frictional force can be calculated using the formula:
F_friction = μ * N
Where:
- μ is the coefficient of friction (0.4 in this case).
- N is the normal force, which for a horizontal surface is equal to the weight of the particle (mass * gravity).
Given that the mass of the particle is 1 kg and the acceleration due to gravity is 10 m/s², we find:
N = mass * gravity = 1 kg * 10 m/s² = 10 N
Now, substituting the values into the frictional force equation:
F_friction = 0.4 * 10 N = 4 N
Step 2: Calculate the Work Done by Friction
The work done by friction as the particle travels a distance of 0.25 m can be calculated using:
W_friction = F_friction * d
Where d is the distance traveled (0.25 m). Thus:
W_friction = 4 N * 0.25 m = 1 J
Step 3: Determine Initial Kinetic Energy
The initial kinetic energy (KE) of the particle can be calculated using the formula:
KE_initial = 0.5 * m * v²
Substituting the values:
KE_initial = 0.5 * 1 kg * (4 m/s)² = 0.5 * 1 * 16 = 8 J
Step 4: Calculate the Kinetic Energy After Friction
After the particle has traveled 0.25 m, the work done by friction reduces the kinetic energy:
KE_final = KE_initial - W_friction
KE_final = 8 J - 1 J = 7 J
Step 5: Determine the Compression of the Spring
When the particle strikes the spring, its kinetic energy will be converted into potential energy stored in the spring. The potential energy (PE) in the spring can be expressed as:
PE_spring = 0.5 * k * x²
Where:
- k is the spring constant (6 N/m).
- x is the compression of the spring.
Setting the final kinetic energy equal to the potential energy of the spring gives us:
7 J = 0.5 * 6 N/m * x²
Solving for x:
7 J = 3 * x²
x² = 7 J / 3 = 2.33
x = √2.33 ≈ 1.53 m
Final Displacement Calculation
The total displacement of the particle from its starting point is the distance it traveled before hitting the spring plus the compression of the spring:
Total Displacement = Distance traveled + Spring compression
Total Displacement = 0.25 m + 1.53 m ≈ 1.78 m
Thus, the final displacement of the particle from its starting point is approximately 1.78 meters.
