Question

Figure 7-21ashows four situations in which a horizontal force acts on the same block, which is initially at rest. The force magnitudes are ${\mathit{F}}_{\mathbf{2}}\mathbf{=}{\mathit{F}}_{\mathbf{4}}\mathbf{=}\mathbf{2}{\mathit{f}}_{\mathbf{1}}\mathbf{=}\mathbf{2}{\mathit{F}}_{\mathbf{3}}$. The horizontal component ${\mathit{v}}_{\mathit{X}}$of the block’s velocity is shown in figure 7-21bfor the four situations. (a) Which plot in figure 7-21b best corresponds to which force in fig. 7-21a? (b) Which plot in figure 7-21c (for kinetic energy K versus time t) best corresponds to which plot in fig. 7-21b ?

Short answer

1. The forces,${\mathrm{F}}_1$,${\mathrm{F}}_2$,${\mathrm{F}}_3$androle="math" localid="1657173006681" ${\mathrm{F}}_4$ corresponds to velocity plot B, A, C, and D respectively.
2. The forces, ${\mathrm{F}}_1$,${\mathrm{F}}_2$ ,${\mathrm{F}}_3$ , and ${\mathrm{F}}_4$ corresponds to kinetic energy plot F, E, F, E respectively.

Step-by-step solution

The given data

1. The plot a-shows four situations of forces acting on the same block which is initially at rest.
2. The magnitudes of the forces are: ${\mathrm{F}}_2={\mathrm{F}}_4=2{\mathrm{F}}_1=2{\mathrm{F}}_3$
3. Plot b- shows four-velocity situations.
4. The plot c-shows four kinetic energy situations.

Understanding the concept of acceleration and kinetic energy

We have to use Newton’s second law to relate force with acceleration. Once we know the acceleration, we can find velocity change for the time by using kinematic equations.

Formulae:

The acceleration of a body,

$\mathbf{a}\mathbf{=}\frac{\mathbf{∆}\mathbf{v}}{\mathbf{t}}$ (1)

The kinetic energy of a body,

$\mathbf{K}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{mv}}^{\mathbf{2}}$ (2)

a) Calculation of the plot of the velocities for the corresponding forces

For force ${\mathrm{F}}_1$

Forceis in the positive x-direction, so acceleration is positive, meaning velocity increases with respect to time from equation (1).

From the given graph of velocity versus time, graph B is for force${\mathrm{F}}_1$.

For force${\mathrm{F}}_2$

Force${\mathrm{F}}_2$is in the positive x-direction, so acceleration is positive, meaning velocity increases with respect to time from equation (1).

From the given graph of velocity versus time, graph A is for force ${\mathrm{F}}_2$because the force${\mathrm{F}}_2$has a greater magnitude than${\mathrm{F}}_1$

For force${\mathrm{F}}_3$

Force ${\mathrm{F}}_3$is in negative x-direction, so acceleration is negative, meaning velocity increases with respect to time in the negative direction from equation (1).

From the given graph of velocity versus time, graph C is for force ${\mathrm{F}}_3$.

For force ${\mathrm{F}}_4$

Force ${\mathrm{F}}_4$is in negative x-direction, so acceleration is negative, meaning velocity increases in the negative x-direction with respect to time from equation (1).

From the given graph of velocity versus time, graph D is for force ${\mathrm{F}}_4$because the magnitude of ${\mathrm{F}}_3$is less than ${\mathrm{F}}_4$

The forces,${\mathrm{F}}_1$, ${\mathrm{F}}_2$, localid="1657174444138" ${\mathrm{F}}_3$and ${\mathrm{F}}_4$corresponds to velocity plot B, A, C, and D respectively.

b) Calculation of the plot of the kinetic energies for the corresponding forces

For force ${\mathrm{F}}_1$

Further, as velocity increases, kinetic energy also increases with time from equation (2), so the graph F is for force ${\mathrm{F}}_1$

For force ${\mathrm{F}}_2$

Further, as velocity increases, kinetic energy also increases with time from equation (2), so the graph E is for force ${\mathrm{F}}_2$because the force ${\mathrm{F}}_2$has a greater magnitude than ${\mathrm{F}}_1$

For force ${\mathrm{F}}_3$

Though the velocity is negative, kinetic energy involves the square of velocity, so the energy increases irrespective of the direction of motion using equation (2). So the graph F is for force ${\mathrm{F}}_3$

For force ${\mathrm{F}}_4$

Though the velocity is negative, kinetic energy involves the square of velocity, so the energy increases irrespective of the direction of motion using equation (ii). So the graph Eis for force ${\mathrm{F}}_4$

The forces, ${\mathrm{F}}_1$,${\mathrm{F}}_2$ , ${\mathrm{F}}_3$and ${\mathrm{F}}_4$corresponds to kinetic energy plot F, E, F, E respectively.