Question 13.1: Billiards, Anyone? Three 0.300-kg billiard balls are placed ...
Billiards, Anyone?
Three 0.300-kg billiard balls are placed on a table at the corners of a right triangle as shown in Figure 13.3. The sides of the triangle are of lengths a = 0.400 m, b = 0.300 m, and c = 0.500 m. Calculate the gravitational force vector on the cue ball (designated m_1) resulting from the other two balls as well as the magnitude and direction of this force.
Conceptualize Notice in Figure 13.3 that the cue ball is attracted to both other balls by the gravitational force. We can see graphically that the net force should point upward and toward the right. We locate our coordinate axes as shown in Figure 13.3, placing our origin at the position of the cue ball.
Categorize This problem involves evaluating the gravitational forces on the cue ball using Equation 13.3. Once these forces are evaluated, it becomes a vector addition problem to find the net force.
\overrightarrow{F}_{12}=-G \frac{m_1 m_2}{r^2} \hat{r}_{12} (13.3)
Analyze Find the force exerted by m_2 on the cue ball:
Find the force exerted by m_3 on the cue ball:
Find the net gravitational force on the cue ball by adding these force vectors:
Find the magnitude of this force:
\begin{aligned} F & =\sqrt{F_{31}{}^2+F_{21}{}^2}=\sqrt{(6.67)^2+(3.75)^2} \times 10^{-11} N \\ & =7.66 \times 10^{-11} N\end{aligned}Find the tangent of the angle θ for the net force vector:
\tan \theta=\frac{F_y}{F_x}=\frac{F_{21}}{F_{31}}=\frac{3.75 \times 10^{-11} N}{6.67 \times 10^{-11} N}=0.562Evaluate the angle θ:
\theta=\tan ^{-1}(0.562)=29.4^{\circ}Finalize The result for F shows that the gravitational forces between everyday objects have extremely small magnitudes.