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 $ \lambda - 3\mu + 7 = 0 $ At time t = 4 seconds, P passes through the point B. Given that $ \lambda = 2 $ b) find the length of AB.

A-Level Edexcel Maths Mechanics Forces: 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 $ \lambda - 3\mu + 7 = 0 $

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Answer

To show that ( \lambda - 3\mu + 7 = 0 ), we need to analyze the forces acting on particle P.

The resultant force ( R ) can be computed as follows:

$R = F_1 + F_2 = (4i - j) + (\lambda i + \mu j) = (4 + \lambda) i + (-1 + \mu) j$

Since the particle moves in the direction of the vector ( (3i + j) ), we can relate the components:

For the i-component: ( 4 + \lambda = k imes 3 ) for some scalar k.
For the j-component: ( -1 + \mu = k imes 1 ) where we say the same scalar ( k ) is used as both forces are in proportion.

From these equations:

From the first equation, we have ( k = \frac{4 + \lambda}{3} )
From the second equation, substituting in k:

$-1 + \mu = \frac{4 + \lambda}{3}$

Rearranging gives:

$\mu = \frac{4 + \lambda}{3} + 1$

We can formulate another expression by eliminating k:

Multiplying both equations by 3:

( 4 + \lambda = 3k )
( -3 + 3\mu = 3k )
Setting them equal gives:

$4 + \lambda + 3 - 3\mu = 0$

Thus, simplifying yields:

$\lambda - 3\mu + 7 = 0$

This proves the required equation.

Step 2

find the length of AB.

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Answer

Given ( \lambda = 2 ), we substitute this value back into the earlier derived equations to find ( \mu ):

From the previous step: $\mu = \frac{4 + 2}{3} + 1 = 2$

Now we can compute the resultant force at ( t = 4 ):

Using ( F_1 ) and ( F_2 ): $R = F_1 + F_2 = (4i - j) + (2i + 2j) = (6i + j) \text{ N}$

Next, we apply Newton's second law: $R = m \cdot a \implies (6i + j) = 4 \cdot a \implies a = \left(\frac{3}{2}, \frac{1}{4}\right)$

Now to find the distance traveled in 4 seconds using the equation of motion: $s = ut + \frac{1}{2} a t^2$ Since the particle starts at rest (u = 0), we have:

= \left(\frac{1}{2} \times \frac{3}{2} \times 16, \frac{1}{2} \times \frac{1}{4} \times 16\right) $$ This simplifies to: $$ s = (12, 2) $$ The position vector for point B is from point A to point B: $$ AB = (12i + 2j) $$ Finally, to find the length of AB: $$ |AB| = \sqrt{(12)^2 + (2)^2} = \sqrt{144 + 4} = \sqrt{148} = 2\sqrt{37} \text{ meters} $$

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 \( \lambda - 3\mu + 7 = 0 \)

find the length of AB.

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