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 Newtons Second Law: 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

96%

114 rated

Answer

To show that the forces result in the motion of P in the direction of the vector (3i + j), we first find the resultant force on P.

The forces in the i and j directions are:

For F₁: 4 N in the i direction and -1 N in the j direction.
For F₂: λ N in the i direction and μ N in the j direction.

So, the resultant force

$F_{ ext{resultant}} = F₁ + F₂ = (4 + λ)i + (-1 + μ)j$

Since P moves in the direction of (3i + j), we can express the relationship between the i and j components of the resultant force and the given motion vector using ratios:

$\frac{4 + λ}{3} = \frac{-1 + μ}{1}$

From the first ratio,

$4 + λ = 3 \Rightarrow λ = 3 - 4 = -1$

From the second ratio,

$-1 + μ = 1 \Rightarrow μ = 1 + 1 = 2$

Now substituting λ and μ values into the equation:

$λ - 3μ + 7 = -1 - 3(2) + 7 = -1 - 6 + 7 = 0$

Thus, we conclude:

$λ - 3μ + 7 = 0$

Step 2

find the length of AB

99%

104 rated

Answer

Given λ = 2, we can now substitute this back into the expression for F₂. We also need to find the resultant force that dictates the motion of P:

$F₂ = (2i + μj)$

Now incorporating the given values:

$F_{ ext{resultant}} = (4 + 2)i + (-1 + μ)j = (6)i + (-1 + μ)j$

To find μ, first apply the previously found result:

Using the equation from part (a),

$λ - 3μ + 7 = 0 \Rightarrow 2 - 3μ + 7 = 0 \Rightarrow 3μ = 9 \Rightarrow μ = 3$

The position at time t = 0 is still at point A, with a resultant force of:

$F_{ ext{resultant}} = (6i + 2j)$

Now, using the second equation of motion with initial conditions (u = 0) and t = 4,

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

Where we need to find the acceleration:

$a = \frac{F_{ ext{resultant}}}{m} = \frac{(6i + 2j)}{4} = (1.5i + 0.5j)$

Calculating the displacement:

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

Therefore, point B from point A is just:

AB = ∥s∥ = $\sqrt{(12)^2 + (4)^2} = \sqrt{144 + 16} = \sqrt{160} = 4\sqrt{10}$ .

Thus the length of AB is:

$4\sqrt{10} ext{ m}$

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

Join the A-Level students using SimpleStudy...