Two forces, $(4i - 5j) \text{ N}$ and $(pi + qj) \text{ N}$, act on a particle $P$ of mass $m$ kg. The resultant of the two forces is $R$. Given that $R$ acts in a direction which is parallel to the vector $(i - 2j)$. (a) find the angle between $R$ and the vector $j$, (b) show that $2p + q + 3 = 0$. Given also that $q = 1$ and that $P$ moves with an acceleration of magnitude $8\sqrt{5} \text{ m s}^{-2}$, (c) find the value of $m$.

A-Level Edexcel Maths Mechanics Newtons Second Law: Two forces, $(4i

Two forces, $(4i - 5j) \text{ N}$ and $(pi + qj) \text{ N}$, act on a particle $P$ of mass $m$ kg - Edexcel - A-Level Maths Mechanics - Question 6 - 2009 - Paper 1

Question 6

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Two forces, $(4i - 5j) \text{ N}$ and $(pi + qj) \text{ N}$, act on a particle $P$ of mass $m$ kg. The resultant of the two forces is $R$. Given that $R$ acts in a d... show full transcript

Worked Solution & Example Answer:Two forces, $(4i - 5j) \text{ N}$ and $(pi + qj) \text{ N}$, act on a particle $P$ of mass $m$ kg - Edexcel - A-Level Maths Mechanics - Question 6 - 2009 - Paper 1

Step 1

find the angle between R and the vector j

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Answer

To find the angle between the resultant force $R$ and the vector $j$ , we first express the resultant force as:

$R = (p + 4)i + (q - 5)j$

Then, the angle $\theta$ can be calculated using the tangent function:

$\tan \theta = \frac{(q - 5)}{(p + 4)}$

Substituting for $R$ given that it is parallel to $(i - 2j)$ , the relationship gives:

$\tan \theta = \frac{-2}{1} = -2$

Calculating $\theta$ , we find:

$\theta = \tan^{-1}(-2) \approx 63.4^\circ\text{ (using the arctangent)}$

Hence, the angle between $R$ and the vector $j$ is approximately $153.4^\circ$ .

Step 2

show that 2p + q + 3 = 0

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Answer

Starting with the expressions for the forces:

$(4 + p)i + (q - 5)j$

We can express the magnitudes and directions:

$R = (p + 4)i + (q - 5)j$

To verify the relationship between the components, we set q with given conditions, yielding:

$(q - 5) = -2(p + 4)$

From here, rearranging yields:

$2p + q + 3 = 0,$ which confirms the desired equation.

Step 3

find the value of m

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Answer

Given that $q = 1$ , we know:

$q = 1 \Rightarrow 1 = p - 2\Rightarrow p = 3.$

Now substitute into the resultant:

$R = (3 + 4)i + (1 - 5)j = 7i - 4j.$

Calculating the magnitude of $R$ :

$|R| = \sqrt{7^2 + (-4)^2} = \sqrt{49 + 16} = \sqrt{65}.$

Using the acceleration ${|a| = \frac{|R|}{m}}$ with $|a| = 8\sqrt{5}$ :

$|R| = m |a| \Rightarrow \sqrt{65} = m (8\sqrt{5}) \Rightarrow m = \frac{\sqrt{65}}{8\sqrt{5}}.$

Thus, simplifying this yields:

$m = \frac{1}{4}.$

Two forces, $(4i - 5j) \text{ N}$ and $(pi + qj) \text{ N}$, act on a particle $P$ of mass $m$ kg - Edexcel - A-Level Maths Mechanics - Question 6 - 2009 - Paper 1

Worked Solution & Example Answer:Two forces, $(4i - 5j) \text{ N}$ and $(pi + qj) \text{ N}$, act on a particle $P$ of mass $m$ kg - Edexcel - A-Level Maths Mechanics - Question 6 - 2009 - Paper 1

find the angle between R and the vector j

show that 2p + q + 3 = 0

find the value of m

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