Question

Figure 15-29gives, for three situations, the displacements of a pair of simple harmonic oscillators (A and B) that are identical except for phase. For each pair, what phase shift (in radians and in degrees) is needed to shift the curve for A to coincide with the curve for B? Of the many possible answers, choose the shift with the smallest absolute magnitude.

Short answer

a) Phase shift needed for A to coincide with B for curve (a) is $\mathrm{\pi }$rad or $180°$.

b) Phase shift needed for A to coincide with B for curve (b) is $\frac{\mathrm{\pi }}{2}$or $90°$.

c) Phase shift needed for A to coincide with B for curve (c) is $-\frac{\mathrm{\pi }}{2}$rad or $-90°$.

Step-by-step solution

The given data 

From the problem figure 15-29 is the graphs of x versus t.

Understanding the concept of the phase shift

We observe the given graphs and we can see the positions of curve A for B. Then we can determine by how many angles should the curve needs to shift to coincide with B.

Calculation of phase shift for curve (a)

a)

In graph (a) A and B have opposite phase, so for curve A needs to shift by $\mathrm{\pi }$rador$180°$ to coincide with B.

Calculation of phase shift for curve (b)

b)

In graph (b) curve A lags behind curve B by $\frac{\mathrm{\pi }}{2}$radians, therefore curve A needs to shift by $\frac{\mathrm{\pi }}{2}$rad or$90°$ to coincide with B.

Calculation of phase shift for curve (c)

c)

In graph (c) curve A leads curve B by $\frac{\mathrm{\pi }}{2}$rad, therefore curve A needs to shift by $-\frac{\mathrm{\pi }}{2}$rador$-90°$to coincide with B.