Question

ball a is attached to one end of a rigid massless rod, while an identical ball b is attached to the center of the rod, as shown in the figure. each ball has a mass of m = 0.530 kg, and the length of each half of the rod is l = 0.320 m. this arrangement is held by the empty end and is whirled around in a horizontal circle at a constant rate, so that each ball is in uniform circular motion. ball a travels at a constant speed of va = 5.40 m/s. find (a) the tension of the part between a and b of the rod and (b) the tension of the part between b and the empty end.

Explanation:

Step1: Find the centripetal - force formula

The centripetal - force formula is $F = \frac{mv^{2}}{r}$, where $m$ is the mass, $v$ is the speed, and $r$ is the radius of the circular path.

Step2: Calculate the tension between A and B ($T_A$)

For ball A, the centripetal force is provided by the tension in the rod between A and B. The radius of the circular path of ball A is $r_A=2L$, and $m = 0.530\ kg$, $v_A = 5.40\ m/s$, $L = 0.320\ m$. Using the centripetal - force formula $F = \frac{mv^{2}}{r}$, we have $T_A=\frac{mv_A^{2}}{2L}$. Substitute the values: $m = 0.530\ kg$, $v_A = 5.40\ m/s$, and $L = 0.320\ m$ into the formula.

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Step3: Calculate the speed of ball B ($v_B$)

Since the angular speed $\omega$ is the same for both balls, and $\omega=\frac{v}{r}$. For ball A, $\omega=\frac{v_A}{2L}$, and for ball B, $v_B=\omega\times L$. Substituting $\omega=\frac{v_A}{2L}$ into the formula for $v_B$, we get $v_B=\frac{v_A}{2}$. Given $v_A = 5.40\ m/s$, then $v_B = 2.70\ m/s$.

Step4: Calculate the tension between B and the empty end ($T_B$)

The centripetal force for ball B is provided by the net - force acting on it. The net - force on ball B is $T_B - T_A$, and the centripetal force for ball B is $F_B=\frac{mv_B^{2}}{L}$. So $T_B - T_A=\frac{mv_B^{2}}{L}$, and $T_B=\frac{mv_B^{2}}{L}+T_A$. Substitute $m = 0.530\ kg$, $v_B = 2.70\ m/s$, $L = 0.320\ m$, and $T_A\approx24.15\ N$ into the formula.

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$T_B=12.07 + 24.15=36.22\ N$

Answer:

(a) $24.15\ N$
(b) $36.22\ N$