Question

1. a 200 n child sitting on a playground swing is being pushed by her father. when the rope makes an angle of 27° to the vertical:

a) what is the force exerted by her father?
b) what is the tension in the rope, t?

Explanation:

Step1: Analyze vertical - force equilibrium

In the vertical direction, the vertical component of the tension $T$ balances the weight of the child. So $T\cos\theta = F_g$, where $F_g = 200\ N$ and $\theta=27^{\circ}$.

Step2: Solve for tension $T$

$T=\frac{F_g}{\cos\theta}=\frac{200}{\cos(27^{\circ})}\approx\frac{200}{0.891}\approx224\ N$.

Step3: Analyze horizontal - force equilibrium

In the horizontal direction, the horizontal component of the tension balances the force exerted by the father. So $F = T\sin\theta$.

Step4: Calculate the force exerted by the father

Since $T\approx224\ N$ and $\theta = 27^{\circ}$, then $F=224\times\sin(27^{\circ})\approx224\times0.454\approx102\ N$.

Answer:

a) The force exerted by her father is approximately $102\ N$.
b) The tension in the rope is approximately $224\ N$.