Question

The velocity$\mathit{v}\mathbf{\left(}\mathit{t}\mathbf{\right)}$of a particle undergoing SHM is graphed in Fig. $\mathbf{15}\mathbf{-}\mathbf{20}\mathbf{b}$. Is the particle momentarily stationary, headed toward$\mathbf{+}{\mathbf{x}}_{\mathbf{m}}$, or headed toward$\mathbf{-}{\mathbf{x}}_{\mathbf{m}}$at (a) point A on the graph and (b) point B? Is the particle at$\mathbf{-}{\mathbf{x}}_{\mathbf{m}}$, at$\mathbf{+}{\mathbf{x}}_{\mathbf{m}}$, at 0, between and 0, or between 0 andlocalid="1657280889199" $\mathbf{+}{\mathbf{x}}_{\mathbf{m}}$when its velocity is represented by (c) point A and (d) point B? Is the speed of the particle increasing or decreasing at (e) point A and (f) point B?

Short answer

1. The particle is headed towards $-{\mathrm{x}}_m$ at point A.
2. The particle is headed towards $+{\mathrm{x}}_m$at point B.
3. Position of particle is between$-{\mathrm{x}}_m$ when velocity is represented by point A.
4. Position of particle is between$-{\mathrm{x}}_m$ when velocity is represented by point B.
5. Speed of particle is decreasing at point A.
6. Speed of particle is increasing at point B.

Step-by-step solution

The given data 

From the problem the figure 15-20(b) is the graph of is given.

Understanding the concept of the SHM

Observing the graph we can find the direction of motion of the particle in the instant. From the absolute value of the velocity, we can find the speed of the particle whether it is increasing or decreasing.

Calculation of the behavior of the particle at point A on the graph

a)

From the graph the maximum velocity of the particle is at point 2 which is at left mean position and point 4 which is at right mean position. At the extreme position, the velocity of particle is 0.

From the graph, the particle is moving from point 2 to point 3 that means the particle is moving from left mean position to left extreme position so the heading of particle at point A is towards $-{\mathrm{x}}_m$.

Calculation of the behavior of the particle at point B on the graph

b)

From the graph we can see that particle is moving from point 3 to point 4 that is from left extreme position to right mean position, so the heading of particle at point B is towards $+{\mathrm{x}}_m$.

Calculation of the position of the particle when velocity is represented by point A

c)

Fromthegraph the particle is moving from point 2 to point 3 that meantheparticle is moving from left mean position to left extreme position so the position of the particle is between$0\mathrm{and}-x_m$ when velocity is represented by point A.

Therefore, the position of the particle is betweenrole="math" localid="1657281422664" $-x_m\mathrm{and}0$ when velocity is represented by point A.

Calculation of the position of the particle when velocity is represented by point B

d)

From the graph, we can see that particle is moving from point 3 to point 4 that is from left mean position to right extreme position, so the position of the particle is between$0\mathrm{and}-{\mathrm{x}}_m$ when the velocity is represented by point B.

Therefore, the position of the particle is betweenrole="math" localid="1657281563133" $-{\mathrm{x}}_m\mathrm{and}0$ when velocity is represented by point B.

Calculation of the behavior of the speed of the particle at point A

e)

From the graph speed of particle is decreasing at point A because absolute value of velocity is decreasing at point A.

Therefore, the speed of particle is decreasing at point A.

Calculation of the behavior of the speed of the particle at point B

f)

From the graph the speed of the particle is increasing because the absolute value of velocity is increasing towards$+{\mathrm{V}}_{max}$.

Therefore, the speed of the particle is increasing at point B.