An object is hanging by a string from your rearview mirror. While you are decelerating at a constant rate from 25 m/s to rest in 6.0 s, (a) what angle does the string make with the vertical, and (b) is it toward the windshield or away from it? [Hint: See Example 4–15.]

Pseudo force may be defined as the kind of force that is not real but caused by a force that is real. Usually, the direction of this force is opposite to the direction of the applied force.

Given data:

The initial velocity is$u=25\mathrm{m}/\mathrm{s}$.

The final velocity is$v=0\mathrm{m}/\mathrm{s}$.

The time is $t=6.0\mathrm{s}$.

(a)

The free-body diagram of the object showing all the real forces can be drawn as follows:

Here, $\theta$is the angle made by the string with the vertical, $a_x$is the acceleration of the vehicle, $F_T$is the tension force, and $F_g$is the gravitational force on the hanging object.

The horizontal acceleration of the object can be calculated as

$\begin{array}{l}{a}_x=\frac{v-u}{t}\\ a_x=\frac{\left(0\mathrm{m}/\mathrm{s}\right)-\left(25\mathrm{m}/\mathrm{s}\right)}{\left(6.0/\mathrm{s}\right)}\\ a_x=-4.167\mathrm{m}{/\mathrm{s}}^2\end{array}$.

Here, a negative sign indicates that the object is decelerating.

Applying equilibrium conditions along the x-direction,

role="math" localid="1645435175732" $\begin{array}{c}\sum F_x=m{a}_x\\ F_T\sin \theta =m{a}_x\dots \left(i\right)\end{array}$

The expression for the force of gravity is given by

$F_g=mg$.

Applying equilibrium conditions along the y-direction,

$\begin{array}{c}\sum F_y=0\\ F_T\cos \theta -F_g=0\\ F_T\cos \theta -mg=0\\ F_T=\frac{mg}{\cos \theta }\end{array}$

Substituting this value in equation (i),

$\begin{array}{c}\left(\frac{mg}{\cos \theta }\right)\sin \theta =m{a}_x\\ mg\tan \theta =m{a}_x\\ \tan \theta =\frac{a_x}{g}\end{array}$

Substituting the values in the above expression,

$\begin{array}{c}\tan \theta =\frac{\left(4.167\mathrm{m}{/\mathrm{s}}^2\right)}{\left(9.8\mathrm{m}{/\mathrm{s}}^2\right)}\\ \theta =23°\end{array}$

Thus, the angle made by the string with the vertical is $23°$.

(b)

As the vehicle is decelerating, the direction of acceleration will be opposite to the vehicle's motion.

A force in the direction of motion is needed to accelerate an object in an accelerating vehicle along with it. An imaginary force acts in the opposite direction of the motion of the vehicle to place the object together at rest. This imaginary force is called the pseudo force.

Due to the pseudo force that acts in the opposite direction of acceleration, the hanging object moves opposite to the direction of the acceleration. Thus, the hanging object moves toward the windshield as the car is decelerating (acceleration is opposite to the motion).