A particle P of mass 4 kg is at rest at the point A on a smooth horizontal plane. At time t = 0, two forces, F1 = (4i - j) N and F2 = (λi + μj) N, where λ and μ are constants, are applied to P. Given that P moves in the direction of the vector (3i + j) (a) show that λ - 3μ + 7 = 0 At time t = 4 seconds, P passes through the point B. Given that λ = 2 (b) find the length of AB.

A-Level Edexcel Maths Mechanics Working with Vectors: A particle P of mass 4 kg is at

A particle P of mass 4 kg is at rest at the point A on a smooth horizontal plane - Edexcel - A-Level Maths Mechanics - Question 3 - 2022 - Paper 1

Question 3

A particle P of mass 4 kg is at rest at the point A on a smooth horizontal plane. At time t = 0, two forces, F1 = (4i - j) N and F2 = (λi + μj) N, where λ and μ are... show full transcript

Worked Solution & Example Answer:A particle P of mass 4 kg is at rest at the point A on a smooth horizontal plane - Edexcel - A-Level Maths Mechanics - Question 3 - 2022 - Paper 1

Step 1

show that λ - 3μ + 7 = 0

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Answer

To determine the equation involving λ and μ, we first calculate the resultant force acting on particle P.

The resultant force, F, can be found by adding the forces:

$F = F_1 + F_2 = (4i - j) + (λi + μj) = (4 + λ)i + (-1 + μ)j$

Next, since P moves in the direction of the vector (3i + j), we can set up a ratio of the components:

The direction ratios give us the relation:

$\frac{(4 + λ)}{3} = \frac{(-1 + μ)}{1}$

From this ratio, we can derive two equations:

From the i component:

$4 + λ = 3k ext{ (for some scalar } k)$

From the j component:

$-1 + μ = k$

Setting these equations equal by eliminating k, we get:

From the first equation:

$k = \frac{λ + 4}{3}$

Substituting into the second equation:

$-1 + μ = \frac{λ + 4}{3}$

Multiplying through by 3 to eliminate the fraction gives:

$-3 + 3μ = λ + 4$

Rearranging leads us to:

$λ - 3μ + 7 = 0$

Step 2

find the length of AB

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Answer

Given that λ = 2, we can substitute this value into our established equation from part (a):

$2 - 3μ + 7 = 0$

Solving for μ gives:

$-3μ + 9 = 0 \Rightarrow 3μ = 9 \Rightarrow μ = 3$

Now, we can calculate the resultant force once again using:

$F_2 = (2i + 3j) N$

Thus, the total force F acting on P is:

$F = (4i - j) + (2i + 3j) = (4 + 2)i + (-1 + 3)j = 6i + 2j$

To find the acceleration a of the particle, we use Newton’s second law:

$F = ma$

Substituting the ratio into the formula:

$a = \frac{F}{m} = \frac{(6i + 2j)}{4} = (1.5i + 0.5j) ext{ m/s}^2$

To find the position of particle P at time t = 4 seconds, we use the equation:

$s = ut + \frac{1}{2}at^2$

Since the initial velocity u = 0, we have:

$s = 0 + \frac{1}{2}(1.5i + 0.5j)(4^2) = \frac{1}{2}(1.5i + 0.5j)(16) = (12i + 4j) ext{ m}$

The position A is at the origin (0,0) and position B is (12, 4). Thus, the length of AB is calculated using the distance formula:

$|AB| = \sqrt{(12 - 0)^2 + (4 - 0)^2} = \sqrt{144 + 16} = \sqrt{160} = 4\sqrt{10} ext{ m}$

A particle P of mass 4 kg is at rest at the point A on a smooth horizontal plane - Edexcel - A-Level Maths Mechanics - Question 3 - 2022 - Paper 1

Worked Solution & Example Answer:A particle P of mass 4 kg is at rest at the point A on a smooth horizontal plane - Edexcel - A-Level Maths Mechanics - Question 3 - 2022 - Paper 1

show that λ - 3μ + 7 = 0

find the length of AB

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