Particles of weight 3 N, 1 N, 4 N, and 2 N are placed at the points $(p, q)$, $(1, p)$, $(q, 4)$, and $(0, 3)$ respectively. The co-ordinates of the centre of gravity of the system are $(-0.5, 1.5)$. Find (i) the value of $p$ (ii) the value of $q$. (b) A uniform triangular lamina in the shape of an isosceles triangle has the vertices $A, B$ and $C$. $|AB| = |BC| = 25 cm$ and $|AC| = 14 cm$. $|BD|$ is the perpendicular height of the triangle, where $D$ is the midpoint of $AC$. Two uniform circles, $S_1$ and $S_2$, are removed from the triangular lamina. $S_1$ has a radius of 3 cm and its center $F_1$ is on the midpoint of $BD$. $S_2$ has a radius of 2 cm and center $E$ on $BD$, where $|DE| = 6 cm$. (i) Taking the coordinates of the vertex $A$ to be $(0,0)$, write down the coordinates of the points $B, C, E$ and $F$. (ii) Calculate the coordinates of the centre of gravity of the remaining shape.

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Particles of weight 3 N, 1 N, 4 N, and 2 N are placed at the points $(p, q)$, $(1, p)$, $(q, 4)$, and $(0, 3)$ respectively - Leaving Cert Applied Maths - Question 6 - 2021

Question 6

Particles of weight 3 N, 1 N, 4 N, and 2 N are placed at the points $(p, q)$, $(1, p)$, $(q, 4)$, and $(0, 3)$ respectively. The co-ordinates of the centre of gravi... show full transcript

Worked Solution & Example Answer:Particles of weight 3 N, 1 N, 4 N, and 2 N are placed at the points $(p, q)$, $(1, p)$, $(q, 4)$, and $(0, 3)$ respectively - Leaving Cert Applied Maths - Question 6 - 2021

Step 1

Find the value of p

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Answer

To find the value of $p$ , we can calculate the X-coordinates of the center of gravity using the formula:

$x_{cg} = \frac{\sum (weight_i \cdot x_i)}{\sum weight_i}$

In this case, the weights are given as follows: 3N, 1N, 4N, and 2N at coordinates $(p, q)$ , $(1, p)$ , $(q, 4)$ , and $(0, 3)$ . The total weight is:

$3 + 1 + 4 + 2 = 10 N$

Setting up the equation for the X-coordinate:

$-0.5 = \frac{3p + 1 \cdot 1 + 4q + 0}{10}$

Simplifying this gives:

$-5 = 3p + 1 + 4q$ $3p + 4q = -6 \\ [1]$

Now, solving for Y-coordinate similarly:

$1.5 = \frac{3q + 1p + 4 \cdot 3}{10}$

This simplifies to:

$15 = 3q + p + 12$ $3q + p = 3 \\ [2]$

Now, we solve the system of equations [1] and [2]:

From [2], we can express $p = 3 - 3q$ and substitute it into [1]:

$3(3 - 3q) + 4q = -6$

This simplifies to: $9 - 9q + 4q = -6$ $-5q = -15$ $q = 3$

Substituting back to find $p$ : $p = 3 - 3(3) \\ p = 2$

Step 2

Find the value of q

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Answer

We previously found that $p = 2$ . Now substituting $p$ into equation [2]:

$3q + 2 = 3$

$3q = 1$ $q = -\frac{1}{3}$

Thus, the values are:

$p = 2$
$q = -3$ .

Step 3

Find Coordinates of Points B, C, E and F

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Answer

Assuming point A at $(0, 0)$ , we first calculate the coordinates of point B. Since $|AB| = 25 cm$ and $|AC| = 14 cm$ , we can use the Pythagorean theorem for $BD$ :

$B(7, 24)$ (based on the given dimensions)
$C(14, 0)$
Midpoint $D$ will be $rac{(0 + 14)}{2}, rac{0 + 24}{2} = (7, 12)$ .
Let $F ext{ at } (7, 24)$ by midpoint of $BD$ .
$E(7, 12)$ being related to D.

Step 4

Calculate the coordinates of the center of gravity of the remaining shape

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Answer

To calculate the center of gravity of the remaining shape after removing circles $S_1$ and $S_2$ , we first compute the areas:

Area of triangle $ABC = \frac{1}{2} \times 14 \times 24 = 168$ .
Area of $S_1 = \pi r_1^2 = \pi (3^2) = 9\pi$ .
Area of $S_2 = \pi r_2^2 = \pi (2^2) = 4\pi$ .

Thus, the area remaining is: $168 - 9\pi - 4\pi$

Next, we compute center of gravity:

Center of gravity before cutting out circles is at $(7, 31)$ when height is 12 cm, for overall geometry taking into account: $7.31 = \left(\frac{168 - 13\pi}{168 - 9\pi}\right)$ .

Particles of weight 3 N, 1 N, 4 N, and 2 N are placed at the points $(p, q)$, $(1, p)$, $(q, 4)$, and $(0, 3)$ respectively - Leaving Cert Applied Maths - Question 6 - 2021

Worked Solution & Example Answer:Particles of weight 3 N, 1 N, 4 N, and 2 N are placed at the points $(p, q)$, $(1, p)$, $(q, 4)$, and $(0, 3)$ respectively - Leaving Cert Applied Maths - Question 6 - 2021

Find the value of p

Find the value of q

Find Coordinates of Points B, C, E and F

Calculate the coordinates of the center of gravity of the remaining shape

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