Statics: Cylinder & Spring system

[spacecataz]

Homework Statement

The linear spring exerts a force at each of its ends that is proportional to the amount of stretch it undergoes. The spring modulus (proportionality constant) is 2 N/cm and its natural (unstretched) length is 1.5 m. Find the normal and friction forces (a) between cylinders A and B and (b) between B and the ground, if the weight of A is 500 N and that of each of B and C is 200 N.

Homework Equations

Sum of moments = 0
Sum of forces = 0
fs = kx

The Attempt at a Solution

$$\Sigma$$MAB = N(r/2) - fr(1+$$\sqrt{3}$$/2) + fs(r/2)
$$\Sigma$$MBGround = (mg-N')(r/2) + f'r(1+$$\sqrt{3}$$/2) + fsr

Am I on the right track? I don't really know where to go from here. Please help! Thanks.

 

[anathema]

Thank you for your question. I am a scientist and I would be happy to assist you with finding the normal and friction forces between the cylinders and the ground.

First, let's define some variables:
- N: normal force between cylinders A and B
- f: friction force between cylinders A and B
- fs: spring force between cylinders A and B
- f': friction force between cylinder B and the ground
- N': normal force between cylinder B and the ground
- r: radius of cylinders A and B
- m: mass of cylinders A and B
- g: acceleration due to gravity
- k: spring modulus

To solve this problem, we will use the equations you have already listed:
- Sum of moments = 0
- Sum of forces = 0
- fs = kx, where x is the amount of stretch of the spring

For the first part (a), we will start with the sum of moments equation:
\SigmaMAB = N(r/2) - fr(1+\sqrt{3}/2) + fs(r/2) = 0

Since the cylinders are in static equilibrium, the sum of moments must be equal to zero. This means that the clockwise and counterclockwise moments must balance each other out. We can use this equation to solve for the unknowns.

Next, we will use the sum of forces equation:
\SigmaFAB = N - mg - fs = 0

Again, since the cylinders are in static equilibrium, the sum of forces must be equal to zero. This means that the forces acting on the cylinders must balance each other out. We can use this equation to solve for N and fs.

For the second part (b), we will use the same equations but with different variables:
\SigmaMBGround = (mg-N')(r/2) + f'r(1+\sqrt{3}/2) + fsr = 0

and
\SigmaFBGround = N' - mg - f' = 0

We can use these equations to solve for N' and f'.

I hope this helps guide you in the right direction. If you have any further questions or need clarification, please let me know.

Scientist

 

[Moetasim]

I would recommend starting by breaking down the problem into smaller, more manageable parts. First, let's consider the forces acting on cylinder A. We know that the weight of A is 500 N, and the spring attached to it exerts a force that is proportional to the amount of stretch it undergoes. This means that the spring force, fs, can be calculated using the equation fs = kx, where k is the spring modulus and x is the amount of stretch. We also know that the natural length of the spring is 1.5 m, so we can use this information to calculate the amount of stretch.

Next, we can consider the forces acting on cylinder B. We know that the weight of B is 200 N, and that there is a normal force, N, acting on it from cylinder A. We can use this information to calculate the normal force, N, using the equation \SigmaF = 0, which means that the sum of all the forces acting on B must equal zero. We can also consider the friction force, f, which is acting on cylinder B due to the contact with cylinder A. We can use the equation \SigmaM = 0 to calculate this force, since we know the distance between the point of contact and the center of mass of cylinder B.

For part (b), we can use similar methods to calculate the normal and friction forces acting on cylinder C due to its contact with the ground.

Overall, it is important to break down the problem into smaller, more manageable parts and use the relevant equations to calculate the forces acting on each component of the system. This will help you arrive at an accurate and well-supported solution.