Chapter 4, Problem 29P

(a)

To determine

The angle that the string makes with the vertical.

Answer to Problem 29P

Solution:

  $23.3°$

Explanation of Solution

Given:

Initial velocity $=V_i=25\text{m/s}$

Final velocity $=V_f=0\text{m/s}$

Time taken $=t=6\mathrm{s}$

Tension force = T

Weight $=mg$

Acceleration $=a$

Formula Used:

The first equation of motion is:

  $\begin{array}{l}{V}_f=V_i+at\\ \Rightarrow a=\frac{V_f-V_i}{t}\end{array}$

Where, $V_i$ is initial velocity, $V_f$ is the final velocity, a is the acceleration and t is the time.

From Newton’s second law of motion:

  $F=ma$

Where, m is the mass and a is the acceleration.

Calculation:

Consider the car moving towards the right and slowing down. The free body diagram for the object hanging is given below. The tension in the string can be represented in terms of its components in horizontal and vertical direction.

  

Acceleration is given as

  $a=\frac{V_f-V_i}{t}$

Inserting the values

  $a=\frac{0-25}{6}=-4.2{\text{m/s}}^{\text{2}}$

In vertical direction:

  $T\text{cos}\theta =mg$

In horizontal direction

  $T\text{sin}\theta =ma$

Take the ratio of the two components and solve for $\theta$ .

  $\begin{array}{l}\frac{T\text{sin}\theta }{T\text{cos}\theta }=\frac{ma}{mg}\\ \tan \theta =\frac{a}{g}\\ \tan \theta =\frac{4.2}{9.8}=0.43\\ \theta ={\tan }^{-1}(0.43)\\ \theta =23.3°\end{array}$

Conclusion:

The angle comes out to be $\theta =23.3°$ .

b)

To determine

Whether the angle is towards windshield or away.

Answer to Problem 29P

Solution:

Towards windshield.

Explanation of Solution

The angle that the string makes with the vertical is $\theta =23.3°$ .

Free body diagram of the given situation:

  

When the car decelerates, the string would move towards the windshield.

Conclusion:

Hence, the object moves towards the windshield.