Answer:
The weight of the pumpkin is 46.793 newtons.
Explanation:
We consider the string-pumpkin system as our system, in which weight of pumpkin create two distinct tensions through cords. The equations of equilibrium are now described:
[tex]\Sigma F_{x} = -T_{1}\cdot \sin \theta_{1}+T_{2}\cdot \sin \theta_{2} = 0[/tex] (Eq. 1)
[tex]\Sigma F_{y} = T_{1}\cdot \cos \theta_{1}+T_{2}\cdot \cos \theta_{2} -W = 0[/tex] (Eq. 2)
Where:
[tex]T_{1}[/tex], [tex]T_{2}[/tex] - Tension through cords, measured in Newtons.
[tex]W[/tex] - Weight of the pumpkin, measured in Newtons.
[tex]\theta_{1}[/tex], [tex]\theta_{2}[/tex] - Inclination of each cord with respect to the vertical, measured in sexagesimal degrees.
If we know that [tex]T_{1} = 55\,N[/tex], [tex]T_{2} = 58\,N[/tex], [tex]\theta_{1} = \theta[/tex] and [tex]\theta_{2} = 121^{\circ}-\theta[/tex], the system of equations becomes:
[tex]-55\cdot \sin \theta +58\cdot \sin (121^{\circ}-\theta) = 0[/tex] (Eq. 1b)
[tex]55\cdot \cos \theta + 58\cdot \cos (121^{\circ}-\theta) = W[/tex] (Eq. 2b)
From Trigonometry, we remember the following identity:
[tex]\sin (121^{\circ}-\theta) = \sin 121^{\circ}\cdot \cos \theta -\cos 121^{\circ}\cdot \sin \theta[/tex]
[tex]\sin (121^{\circ}-\theta) = 0.857\cdot \cos \theta -0.515\cdot \sin \theta[/tex] (Eq. 3)
By applying (Eq. 3) in (Eq. 1b), we get the following expression after expanding and simplifying algebraically:
[tex]-55\cdot \sin \theta + 58\cdot (0.857\cdot \cos \theta - 0.515\cdot \sin \theta) = 0[/tex]
[tex]-84.87\cdot \sin \theta +49.706\cdot \cos \theta = 0[/tex]
And we solve the equation for [tex]\theta[/tex]:
[tex]\tan \theta = \frac{49.706}{84.87}[/tex]
[tex]\theta = \tan^{-1} 0.585[/tex]
[tex]\theta \approx 30.328^{\circ}[/tex]
Then, the weight of the pumpkin is: ([tex]\theta \approx 30.328^{\circ}[/tex])
[tex]W = 55\cdot \cos 30.328^{\circ}+58\cdot \cos (121^{\circ}-30.328^{\circ})[/tex]
[tex]W = 46.793\,N[/tex]
The weight of the pumpkin is 46.793 newtons.
The weight of the pumpkin hanging from the two scales is 98.36 N.
The given parameters;
The weight of the pumpkin hanging from the two scales is equal to the resultant of the two scale readings.
The resultant of the two force is calculated as follows;
R² = a² + b² - 2abcos(θ)
R² = 55² + 58² - 2(55)(58)cos(121)
R² = 9674.7
[tex]R = \sqrt{9674.7} \\\\R = 98.36 \ N[/tex]
Thus, the weight of the pumpkin hanging from the two scales is 98.36 N.
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