I'm not sure whether this is a geometry or mechanics problem, and I'm not too hot on either so either way I need help 🙂 I think it's pure geometry. Also I am not really sure how to formulate this problem, so again, please bear with me, this is really not my area of expertise.
I have a bar of fixed length B suspended from two strings, of lengths $R_1$ and $R_2$, fixed at their other ends to two fixed points, $P_1$, $P_2$, which lie on a horizontal line, and are a fixed horizontal distance apart, $S$. For convenience I've defined these fixed points as at positions $(0,0)$ and $(S, 0)$. $R_1$ and $R_2$ are not fixed, they can be varied.
I want to calculate what the lengths of the two strings, $R_1$ and $R_2$, need to be in order for the centre of the bar to lie at some specific location, which we will call $(x,y)$
In other word I want a function $F(x,y) \implies (R_1,R_2)$
I understand that there maybe more than one solution.
I have tried to solve this by looking at the formula for the circles centred on $P_1$ and $P_2$, and $(x,y)$ of radii $R_1$ and $R_2$ and $B$, and then attempting the solve these simultaneously; however my maths is just not up to it.
I've also tried to solve this mechanically by analysis the tensions in the strings assuming the bar has some mass $M$, and then working back form the tension to the angles the strings have to make with the bar, and then solve that lot simultaneously to find the string lengths that would give those angles; but again my maths just can't cope.
Can anyone assist me please?
Best Answer
We assume that, by now, you have realized that this problem cannot be solved using geometry alone. As @David.k has pointed out in his comment, the equilibrium of coplanar forces and moments acting on the bar must also be taken into account. However, we only need to bother about the directions and not the magnitude of these quantities.
Now, if your knowledge of Engineering Mechanics is as good as that of mathematics, you have to have blind faith in us, when we state that three coplanar forces acting on a body in equilibrium concur, i.e. meet at a point. For instance, the bar shown in $\mathrm{Fig. 1}$ is not in equilibrium, because there is a counterclockwise moment proportional to the distance $p_x$, which pulls one of its end (i.e. $A$) downwards and pushes its other end (i.e. $B$) upwards. This moment strives to change the position of the bar by adjusting the strings until the point $P$ lands somewhere on the vertical line passing through its center of gravity. When the system attains its stability, it may look like the one shown in $\mathrm{Fig. 2}$
We are going to exploit the above-mentioned phenomenon to concoct a method to determine the length of the two strings to hang a bar of given length ($2b$) from two hooks fixed a known distance $d_1+d_2$ apart, so that the midpoint of the bar coincides with a given point in space, for instance $O$. This method works if and only if lengths $b, d_1, d_2$, and $h$ are greater than zero. The case, in which $d_1=d_2=0$, has to be analysed using common sense.
$\underline{\mathrm{Method}}$
To understand the derivation of this method, you need to polish up your knowledge of Coordinate Geometry. Without loss of generality, we assume that the midpoint $O$ of the bat lies at the origin of the Cartesian coordinate system. As shown in $\mathrm{Fig. 2}$, the two lines representing the strings have the same $y$-intercept, if their point of intersection $P$ lies on the $y$-axis, which is the vertical line passing through the CG of the bar. The only unknown in this system is $\phi$, the bar’s tilt to the horizontal - negative $x$-axis to be precise. Please note that $\phi$ is always seen as a positive quantity and. measured clockwise to avoid ambiguity. Therefore, when we get a negative angle as the answer for $\phi$, we make it positive by subtracting its absolute value from the full angle, i.e. $360^o$.
Using the points $A, B, C$, and $D$ and their coordinates depicted in the figure, we can express the equations of the strings $DA$ and $CB$ as
$$y=\space\space\space\left(\frac{h-b\sin\left(\phi\right)}{b\cos\left(\phi\right)–d_1}\right)x + h + \left(\frac{h-b\sin\left(\phi\right)}{b\cos\left(\phi\right)–d_1}\right)d_1, \tag{String $\it{DA}$}$$
$$y=-\left(\frac{h+b\sin\left(\phi\right)}{b\cos\left(\phi\right)–d_2}\right)x + h + \left(\frac{h+b\sin\left(\phi\right)}{b\cos\left(\phi\right)–d_2}\right)d_2. \tag{String $\it{CB}$}$$
When we equate the two $y$-intercepts of these lines, we get, $$h + \left(\frac{h-b\sin\left(\phi\right)}{b\cos\left(\phi\right)–d_1}\right)d_1 = h + \left(\frac{h+b\sin\left(\phi\right)}{b\cos\left(\phi\right)–d_2}\right)d_2.$$
When we simplify this, we have, $$\Big(h-b\sin\left(\phi\right)\Big)\Big(b\cos\left(\phi\right)–d_2\Big)d_1 = \Big(h+b\sin\left(\phi\right)\Big)\Big(b\cos\left(\phi\right)–d_1\Big)d_2.$$
After eliminating $\cos\left(\phi\right)$from the left-hand side of the above expression, we simplify it further to obtain the following equation of degree four in $\sin\left(\phi\right)$.
$$a_4\sin^4\left(\phi\right) + a_3\sin^3\left(\phi\right) + a_2\sin^2\left(\phi\right) + a_1\sin\left(\phi\right) + a_0 = 0, \quad\mathrm{where,} \tag{1}$$ $a_4 = b^2\left(d_1+d_2\right)^2,\space a_1 = -a_3 = 2hb\left(d_1^2-d_2^2\right),\space a_0 = - h^2\left(d_1-d_2\right)^2,\space$ and $\space a_2 = 4d_1^2d_2^2 - a_4 – a_0$.
To solve this equation to find the values of $\phi$, you have to use numerical methods or use a tool such as Wolfram Mathematica. This equation has either four complex roots or two complex roots and two real roots. In the former case, this problem has no ($real$) solution. In the later instance, we have two mathematically valid solutions, where one of them is always positive and the other is always negative. However, only one of them leads to a system in stable equilibrium. $\mathrm{Fig. 3}$ shows an example, where we provide both mathematically valid solutions. We hope that you can visualize in this diagram why configuration shown to the right is in neutral equilibrium and, thus, making it useless in practice.
Equation (2) gives us the two solutions, type of which we are now familiar with – the bar in stable and neutral equilibrium. Unlike in the other two scenarios, the bar is horizontal in both states of equilibrium (see $\mathrm{Fig. 4}$). Furthermore, when $b \ge d$ (see $\mathrm{Fig. 5}$), as indicated by equation (3), this scenario possesses two additional stable equilibrium states. These configurations of equilibrium are in complete agreement with the laws of mechanics, because the two strings are parallel to $y$-axis and, therefore, all three line meet at infinity as required by those laws. Equation (3) does not have real solutions when $b \lt d$.
We suggest that you should work out the special case, where $b, d_1, h > 0$ and $d_2 = 0$, to test yourself the knowledge you have acquired by reading this answer. As we have already noted, the case of $d_1=d_2=0$ cannot be analysed using this method, because in this particular case we have $a_4 = a_3 = a_2 = a_1 = a_0 = 0$, i.e. equation (1) vanishes. That does not mean that there is no stable configuration when $d_1=d_2=0$. On the contrary, there is a $nice$ solution. We would like to leave it to OP to find that solution.
After choosing the value of $\phi$, which gives the configuration in the state of stable equilibrium, you can use the following equations to determine the lengths of the two strings.
$$s_1 = \sqrt{\Big(b\cos\left(\phi\right)-d_1\Big)^2 + \Big(h-b\sin\left(\phi\right)\Big)^2} \tag{String $\it{DA}$}$$
$$s_2 = \sqrt{\Big(d_2-b\cos\left(\phi\right)\Big)^2 + \Big(h+b\sin\left(\phi\right)\Big)^2} \tag{String $\it{CB}$}$$