Topic: Theory of Machines
Find out the correct freebody diagram from the middle mass for the given spring-mass system in below figure
- A. $2k(X_{1}-X_{2})$ [left], $k(X_{3}-X_{2})$ [right]
- B. $2k(X_{2}-X_{1})$ [left], $k(X_{3}-X_{2})$ [right]
- C. $2k(X_{2}-X_{1})$ [left], $k(X_{2}-X_{3})$ [right]
- D. $2k(X_{1}-X_{2})$ [left], $k(X_{2}-X_{3})$ [right]
Answer: B. $2k(X_{2}-X_{1})$ [left], $k(X_{3}-X_{2})$ [right]
Explanation: When the middle mass ($2m$) is displaced by $X_{2}$, the restoring force from the left spring (stiffness $2k$) depends on its relative elongation $(X_{2}-X_{1})$ and acts to oppose this motion. Similarly, the right spring (stiffness $k$) exerts a force proportional to $(X_{3}-X_{2})$. The correct free-body representation shows these internal restoring forces acting on the middle mass correctly oriented against relative displacements.