Linear Vibrations Lab

Worked examples › Example 2

Example 2Pulley system with a hanging spring

EOMpulleysstatic equilibrium

Problem

Blocks A and B, of masses $m$ and $3m$, are connected by a cable–pulley system. A spring of stiffness $3k$ connects A to the wall, and a spring of stiffness $k$ connects B to the ground. $x$ is the position of A, with both springs unstretched at $x=0$. The horizontal surface is smooth.

Find: (a) the equation of motion in terms of $x$; (b) the natural frequency.

InteractiveThe system in motion

Displacements in the drawing are scaled up so the motion is visible; the plot shows the true values.

x(t)Static equilibrium x_st

Solution

Every solution follows the same five steps: coordinates, free-body diagrams, kinetics, kinematics, solve. Try the problem first, then open one step at a time to check your work.

Step 1 · Choose a coordinate system

Let $x$ be A's position, positive to the right (toward the pulley), with both springs unstretched at $x=0$. Let $x_B$ be B's displacement, positive downward.

Step 2 · Draw the free-body diagrams
Block AAT3kxmgNBlock B + light pulley3mTT3mgk xB
Free-body diagrams of A and of B. The lower spring is compressed by xB, so it pushes B upward.
Step 3 · Kinetics (Newton–Euler equations)

Block A (horizontal): $\;m\ddot x=T-3kx$

Block B (down +): $\;3m\ddot x_B=3mg-2T-kx_B$

Step 4 · Kinematics (constraints)

Cable length is constant and B hangs on two strands, so

$$x_B=\tfrac12x\qquad\Longrightarrow\qquad \ddot x_B=\tfrac12\ddot x$$
Step 5 · Solve

From B: $T=\tfrac12\left(3mg-\tfrac k2x-\tfrac32m\ddot x\right)$. Substitute into A:

$$\tfrac74m\ddot x+\tfrac{13}4kx=\tfrac32mg\qquad\Longleftrightarrow\qquad 7m\ddot x+13kx=6mg$$ $$\omega_n=\sqrt{\frac{13k}{7m}},\qquad x_{st}=\frac{6mg}{13k}$$

Because $x$ is measured from the unstretched position, gravity stays as a constant force; the motion is a vibration about $x_{st}$.

Answer

(a) $\displaystyle 7m\ddot x+13kx=6mg$   (equivalently $\tfrac74m\ddot x+\tfrac{13}4kx=\tfrac32mg$)

(b) $\displaystyle \omega_n=\sqrt{\frac{13k}{7m}}$

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