Question #238611

The leg and cast in the figure below weigh 280 N (w1). Determine the weight w2 and the angle 𝛼 needed so that no force is exerted on the hip joint by the leg plus cast


w2 = ___ N

𝛼=  ___°



1
Expert's answer
2021-09-20T16:26:13-0400

Question:

The leg and cast in the figure below weigh 280N (w1). Determine the weight w2 and the angle 𝛼 needed so that no force is exerted on the hip joint by the leg plus cast.




Solution:

Given:

T3=w1=220NT1=110Nθ1=40°T_3 = w_1 = 220N\\ T_1 = 110 N\\ \theta_1 = 40°


θ2=?T2=w2=?\theta_2 =?\\ T_2 = w_2=?


The representation of the image is;




Newton’s 3rd Law of Motion has already been used to isolate the joint from its physical connections by way of action-reaction pair.


Newton’s 2nd Law of Motion Use components. The x axis is the horizontal and the y axis the vertical directions.


Because there is no horizontal displacement

ΣFx=0T1x+T2x=T1cosθ1+T2cosθ2=0\Sigma F_x = 0\\ –T_{1x} + T_{2x} = -T_1cos\theta_1 + T_2cos\theta_2 = 0


Because Upward force is equal to downward force

ΣFy=0T1y+T2yW1=0T1sinθ1+T2sinθ2W1=0\Sigma Fy = 0 T_{1y} + T_{2y} – W_1 = 0 \\T_1sin\theta_1 + T_2sin\theta_2 – W_1 = 0





T2 and θ2\theta_2 are both unknown. They can be eliminated. These two variables occur together as T2sinθ2T_2sin\theta_2 and T2cosθ2T_2cos\theta_2. This suggests solving for the angle θ2\theta_2 by solving for;


tanθ2=T2sinθ2T2cosθ2=W1T1sinθ1T1cosθ1\tan\theta_2 = \dfrac{T_2\sin\theta_2}{T_2\cos\theta_2}= \dfrac{W_1 -T_1\sin\theta_1}{T_1\cos\theta_1}


Then the angle follows from;


θ2=α=tan1W1T1sinθ1T1cosθ1\theta_2 =\alpha=\tan^{-1} \dfrac{W_1 -T_1\sin\theta_1}{T_1\cos\theta_1}


=tan1220110sin40°T1cos40°=\tan^{-1} \dfrac{220 -110\sin40°}{T_1\cos40°}


=tan1149.384.3=60.6°=\tan^{-1} \dfrac{149.3}{84.3}= 60.6°



T2=w2=T1cosθ1cosθ2=110cos40°cos60.6°T_2 =w_2 = T_1\dfrac{\cos\theta_1}{\cos\theta_2}= 110\dfrac{\cos40°}{\cos60.6°}


=172N= 172N

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Comments

Mike Sibajene
20.05.22, 23:27

Thank you, the working was simplified well

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