In Figure 8-34, a 1.2 kg block is held at rest against a spring with a force constant k = 785 N/m. Initially, the spring is compressed a distance d. When the block is released, it slides across a surface that is frictionless, except for a rough patch of width 5.0 cm that has a coefficient of kinetic friction µk = 0.44. Find d such that the block's speed after crossing the rough patch is 2.0 m/s.
To find the distance d, we need to analyze the forces acting on the block and use the principle of conservation of mechanical energy.
First, let's identify the forces acting on the block:
1. The force applied by the spring, which can be calculated using Hooke's Law: F_spring = -k * x, where k is the force constant and x is the displacement of the spring from its equilibrium position. Since the block is released, x = d.
2. The force of friction acting on the block as it crosses the rough patch. The frictional force can be calculated using the equation: F_friction = µk * N, where µk is the coefficient of kinetic friction and N is the normal force. The normal force is equal to the weight of the block, N = mg, where m is the mass of the block and g is the acceleration due to gravity.
3. The weight of the block, which is equal to mg.
Now, let's set up the conservation of mechanical energy equation:
Initial Potential Energy + Initial Kinetic Energy = Final Potential Energy + Final Kinetic Energy
Since the block is initially held at rest against the spring, the initial kinetic energy is zero. The initial potential energy is given by the potential energy stored in the compressed spring:
Initial Potential Energy = (1/2) * k * (d^2)
The final kinetic energy of the block is given by:
Final Kinetic Energy = (1/2) * m * v^2
where m is the mass of the block and v is the desired velocity of the block after crossing the rough patch.
The final potential energy is zero since the block is not at any height above the ground.
Now, let's solve for d:
(1/2) * k * (d^2) = (1/2) * m * v^2
Since all the variables are given except for d, we can rearrange the equation to solve for d:
d = sqrt((m * v^2) / k)
Plug in the given values:
m = 1.2 kg
v = 2.0 m/s
k = 785 N/m
d = sqrt((1.2 kg * (2.0 m/s)^2) / 785 N/m)
Calculate d using the equation and solve the expression:
d ≈ 0.109 m (rounded to three decimal places)
Therefore, the distance d that the spring needs to be compressed is approximately 0.109 meters to achieve a speed of 2.0 m/s after crossing the rough patch.