Question

A block of mass m = 2.75 kg slides along a horizontal table with speed v0 = 1.00 m/s. At x = 0 it hits a spring with spring constant k = 82.00 N/m and it also begins to experience a friction force. The coefficient of friction is given by μ = 0.100. How far has the spring compressed by the time the block first momentarily comes to rest?

Answer 1

Answer 2

To find the distance the spring has compressed when the block first momentarily comes to rest, we need to analyze the forces acting on the block and use the conservation of mechanical energy.

1. Find the force of friction: The friction force acting on the block can be calculated using the formula: F_friction = μ * F_normal. The normal force (F_normal) is equal to the weight of the block (mg), where g is the acceleration due to gravity (approximately 9.8 m/s^2). So, F_friction = μ * mg.

F_friction = (0.100) * (2.75 kg) * (9.8 m/s^2) F_friction ≈ 2.696 N

2. Calculate the work done by friction: The work done by the friction force can be calculated as W_friction = F_friction * d, where d is the distance the block has traveled. Since the block comes to rest, the work done by friction is equal to the initial kinetic energy of the block: W_friction = 0.5 * m * v0^2.

0.5 * (2.75 kg) * (1.00 m/s)^2 = 2.696 N * d

d ≈ (0.5 * 2.75 * 1.00^2) / 2.696

d ≈ 0.509 m

Therefore, the spring has compressed approximately 0.509 meters (or 50.9 centimeters) by the time the block first momentarily comes to rest.