A block of mass m = 4.20 kg slides along a horizontal table with speed v0 = 9.50 m/s. At x = 0 it hits a spring with spring constant k = 62.00 N/m and it also begins to experience a friction force. The coefficient of friction is given by μ = 0.150. How far has the spring compressed by the time the block first momentarily comes to rest?
To find out how far the spring has compressed when the block first comes to rest, we need to consider both the spring force and the friction force acting on the block.
1. Calculate the force of friction:
The friction force can be calculated using the equation:
f_friction = μ * N
where μ is the coefficient of friction and N is the normal force. In this case, the normal force is equal to the weight of the block, which can be calculated as:
N = m * g
where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s²). Thus, the friction force becomes:
f_friction = μ * m * g
2. Calculate the initial acceleration of the block:
The initial acceleration of the block when it hits the spring can be found using Newton's second law:
F_net = m * a
where F_net is the net force acting on the block. The net force comprises the spring force and the friction force:
F_net = -k * x - f_friction
The negative sign indicates that the spring force and the friction force act in the opposite direction to the motion of the block.
3. Find the compression distance:
When the block comes to rest momentarily, its acceleration will be zero. Therefore, we can set the net force equal to zero and solve for the compression distance (x):
0 = -k * x - f_friction
Rearranging the equation gives:
k * x = -f_friction
Substituting the value of f_friction, we have:
k * x = -μ * m * g
Finally, solve for x:
x = (−μ * m * g) / k
Now, plug in the given values for the mass (m = 4.20 kg), the coefficient of friction (μ = 0.150), the acceleration due to gravity (g ≈ 9.8 m/s²), and the spring constant (k = 62.00 N/m) to calculate the compression distance (x).