4. four forces are exerted on the eyebolt as shown. if the net effect on the bolt is a direct pull of 1200 n…

Explanation:

Step1: Resolve forces in x - direction

Since the net force in the x - direction is zero ($F_{net,x}=0$), we have: $- 720+800\cos30^{\circ}+T\cos\theta - 480\cos\theta=0$. $-720 + 800\times\frac{\sqrt{3}}{2}+(T - 480)\cos\theta=0$. $-720 + 400\sqrt{3}+(T - 480)\cos\theta=0$. $(T - 480)\cos\theta=720 - 400\sqrt{3}$.

Step2: Resolve forces in y - direction

The net force in the y - direction is $F_{net,y}=1200$ N. So, $480\sin\theta+T\sin\theta+800\sin30^{\circ}=1200$. $(480 + T)\sin\theta+800\times\frac{1}{2}=1200$. $(480 + T)\sin\theta=1200 - 400=800$.

Step3: Express $\sin\theta$ and $\cos\theta$ in terms of T

From $(T - 480)\cos\theta=720 - 400\sqrt{3}$, we get $\cos\theta=\frac{720 - 400\sqrt{3}}{T - 480}$. From $(480 + T)\sin\theta=800$, we get $\sin\theta=\frac{800}{480 + T}$. Using the identity $\sin^{2}\theta+\cos^{2}\theta = 1$, we substitute the expressions for $\sin\theta$ and $\cos\theta$: $\left(\frac{800}{480 + T}\right)^{2}+\left(\frac{720 - 400\sqrt{3}}{T - 480}\right)^{2}=1$. After simplification and solving the above - equation (by cross - multiplying and expanding), we first note that $720-400\sqrt{3}\approx720 - 400\times1.732=720 - 692.8 = 27.2$. Let's solve the y - direction equation for $T$ in terms of $\theta$: $T=\frac{800}{\sin\theta}-480$. Substitute into the x - direction equation: $\left(\frac{800}{\sin\theta}-960\right)\cos\theta=27.2$. Another way is to assume $\theta$ and solve iteratively or use a graphing calculator. We can also solve the system of equations: From the x - direction: $T\cos\theta=720 - 400\sqrt{3}+480\cos\theta$. From the y - direction: $T\sin\theta=800 - 480\sin\theta$. Dividing the second equation by the first equation gives: $\tan\theta=\frac{800 - 480\sin\theta}{720 - 400\sqrt{3}+480\cos\theta}$. By trial and error or using a scientific calculator to solve the system of equations: First, from the x - direction: $-720+800\times\frac{\sqrt{3}}{2}+T\cos\theta - 480\cos\theta=0$ simplifies to $T\cos\theta=720 - 400\sqrt{3}+480\cos\theta$. From the y - direction: $480\sin\theta+T\sin\theta + 400=1200$ or $T\sin\theta=800 - 480\sin\theta$. We know that $800\cos30^{\circ}=400\sqrt{3}\approx692.8$. $-720 + 692.8+(T - 480)\cos\theta=0$ gives $(T - 480)\cos\theta=27.2$. $480\sin\theta+T\sin\theta=800$. Let's solve directly: From the x - component: $-720 + 800\times\frac{\sqrt{3}}{2}+T\cos\theta-480\cos\theta = 0$ $ - 720+400\sqrt{3}+(T - 480)\cos\theta=0$ From the y - component: $480\sin\theta+T\sin\theta+800\times\frac{1}{2}=1200$ $(480 + T)\sin\theta=800$ We can rewrite the x - equation as $T\cos\theta=720 - 400\sqrt{3}+480\cos\theta$ and the y - equation as $T\sin\theta=800 - 480\sin\theta$ Dividing $T\sin\theta$ by $T\cos\theta$ gives $\tan\theta=\frac{800 - 480\sin\theta}{720 - 400\sqrt{3}+480\cos\theta}$ By solving the system: First, from x - direction: $T\cos\theta=720 - 400\sqrt{3}+480\cos\theta$ From y - direction: $T\sin\theta=800 - 480\sin\theta$ Squaring and adding: $T^{2}= (720 - 400\sqrt{3}+480\cos\theta)^{2}+(800 - 480\sin\theta)^{2}$ Also, from x - direction: $T=\frac{720 - 400\sqrt{3}+480\cos\theta}{\cos\theta}$ Substitute into y - direction: $(480+\frac{720 - 400\sqrt{3}+480\cos\theta}{\cos\theta})\sin\theta=800$ After simplification: $480\sin\theta\cos\theta+(720 - 400\sqrt{3})\sin\theta+480\sin\theta\cos\theta=800\cos\theta$ $960\sin\theta\cos\theta+(720 - 400\sqrt{3})\sin\theta - 800\cos\theta=0$ Using a scientific calculator to solve for $\theta$ and then for $T$: We find that $\theta = 53.1^{\circ}$ Substitute $\theta$ into the y - direction equation $(480 + T)\sin\theta=800$ $(480 + T)\sin(53.1^{\circ})=800$ $(480 + T)\times0.8=800$ $480+T = 1000$ $T = 520$ N

Answer:

$T = 520$ N, $\theta=53.1^{\circ}$