In the figure a 1.2 block is held at rest against a spring with a force constant = 740 . Initially, the spring is compressed a distance . When the block is released, it slides across a surface that is frictionless except for a rough patch of width 5.0 that has a coefficient of kinetic friction = 0.44.
To find the maximum distance the block slides after being released, we need to consider the forces acting on the block and use the principles of Newton's Laws of Motion.
First, let's analyze the situation. The block is initially held at rest against a spring, so the net force acting on the block is zero. When the block is released, it starts to move due to the force exerted by the spring. As the block slides across the surface, it experiences a frictional force opposing its motion. We need to find the maximum distance the block slides before coming to a stop.
To find the maximum distance, we'll consider two key points: the point where the block first leaves the rough patch and the point where it comes to a stop.
Step 1: Find the force exerted by the spring (Fs):
The force exerted by the spring can be determined using Hooke's Law:
Fs = k * x
where Fs is the force exerted by the spring, k is the force constant, and x is the compression distance of the spring. In this case, x is given as 0.12 m, and k is given as 740 N/m. Substituting these values into the equation:
Fs = 740 N/m * 0.12 m
= 88.8 N
The force exerted by the spring is 88.8 N.
Step 2: Determine the frictional force (Ff):
The frictional force can be calculated using the equation:
Ff = μ * N
where Ff is the frictional force, μ is the coefficient of kinetic friction, and N is the normal force. In this case, the normal force is equal to the weight of the block since there is no vertical acceleration. The weight (W) can be determined using the equation:
W = mg
where m is the mass of the block and g is the acceleration due to gravity. The mass of the block is not given in the question, so we cannot calculate the exact value. However, we can proceed with the general formula and substitute the values later.
Ff = μ * (mg)
Step 3: Calculate the maximum distance (d):
The maximum distance the block slides can be determined by analyzing the forces acting on the block. At the point where the block leaves the rough patch, the force exerted by the spring (Fs) is equal to the frictional force (Ff). Therefore:
Fs = Ff
Plugging in the values we found earlier:
88.8 N = μ * (mg)
Now, we can solve this equation for d, the maximum distance the block slides across the rough patch.
To summarize the steps required:
1. Determine the force exerted by the spring using Hooke's Law (Fs = k * x).
2. Calculate the frictional force using the equation Ff = μ * N.
3. Set the force exerted by the spring equal to the frictional force and solve for d.
Please note that in order to obtain the exact value for the maximum distance, we need to know the mass of the block.