Question

The angle between the gravitational force on a planet by a star and the momentum of the planet is $\mathbf{61}$at a particular instant. At this instant the magnitude of the planet’s momentum is $\mathbf{3}\mathbf{.}\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{29}}\mathbf{kgm}\mathbf{/}\mathbf{s}$, and the magnitude of the gravitational force on the planet is $\mathbf{1}\mathbf{.}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{23}}\mathbf{N}$. (a) What is the parallel component of the force on the planet by the star? (b) What will be the magnitude of the planet’s momentum after $8h$?

Short answer

(a)$8.7\times {10}^{22}N\hat{p}$

(b)$3.15184\times {10}^{29}kgm/s$

Step-by-step solution

Identification of the given data 

The given data is listed below as,

• The angle between the planet’s gravitational force and momentum is, $\theta ={61}^0$
  

• The magnitude of the planet’s initial momentum is,$p_i=3.1\times {10}^{29}kgm/s$

• The magnitude of the planet’s gravitational force is,$F=1.8\times {10}^{23}N$

Significance of the parallel force

The parallel force acts in the opposite or same direction at the different points of a particular object.

The equation of the parallel component of the force is expressed as-

$F=\left|F\right|\cos \theta \hat{p}$ …(1)

Here, $F_{\parallel }$is the parallel force, $\left|F\right|$is the absolute value of the gravitational force, $\hat{p}$is the unit vector and $\theta$is the angle between the momentum and gravitational force.

Determination of the parallel component of the force on the planet

(a)

For, $\left|F\right|=1.8\times {10}^{23}N$and $\theta =0$in equation (1).

$$\begin{gathered} F=\left(1.8\times {10}^{23}N\times \cos 0^{\circ }\right)\hat{p} \\ =\left(1.8\times {10}^{23}N\right)\hat{p} \end{gathered}$$

Thus, the parallel component of the force on the planet by the star is $\left(1.8\times {10}^{23}N\right)\hat{p}$.

Determination of the magnitude of the planet’s final momentum

The equation of the magnitude of the planet’s momentum after $8h$is expressed as,

$p_f=p_i+F_{net}∆t$

Here, $p_f$is the final momentum, $p_i$is the initial momentum, $F_{net}$is the net force exerted by the planet and $∆t$is the difference in the time period.

For$p_i=3.1\times {10}^{29}kgm/s$, $p_i=1.8\times {10}^{23}N$and $∆t=8h-0=8h$.

$$\begin{gathered} p_f=\left(3.1\times {10}^{29}kgm/s\right)+\left(1.8\times {10}^{23}N\right)\times \left(8h\times \frac{3600s}{1h}\right) \\ =\left(3.1\times {10}^{29}kgm/s\right)+5.184\times {10}^{27}N.s\times \frac{1kg.m/s^2}{1N} \\ =3.15184\times {10}^{29}kgm/s \end{gathered}$$

Thus, the magnitude of the planet’s momentum after $8h$is $3.15184\times {10}^{29}kgm/s$.