Two forces P and Q act on a particle at a point O. The force P has magnitude 15 N and the force Q has magnitude X newtons. The angle between P and Q is 150°, as shown in Figure 1. The resultant of P and Q is R. Given that the angle between R and Q is 50°, find (a) the magnitude of R, (b) the value of X.

A-Level Edexcel Maths Mechanics Forces: Two forces P and Q act on a part

Two forces P and Q act on a particle at a point O - Edexcel - A-Level Maths Mechanics - Question 5 - 2008 - Paper 1

Question 5

Two forces P and Q act on a particle at a point O. The force P has magnitude 15 N and the force Q has magnitude X newtons. The angle between P and Q is 150°, as show... show full transcript

Worked Solution & Example Answer:Two forces P and Q act on a particle at a point O - Edexcel - A-Level Maths Mechanics - Question 5 - 2008 - Paper 1

Step 1

(a) the magnitude of R

96%

114 rated

Answer

To find the magnitude of the resultant force R, we can use the sine rule. According to the problem, we have:

$rac{15}{ ext{sin}(50^ ext{o})} = \frac{R}{ ext{sin}(30^ ext{o})}$

Rearranging gives:

$R = 15 \cdot \frac{\text{sin}(30^ ext{o})}{\text{sin}(50^ ext{o})}$

Calculating the values:

$\text{sin}(30^ ext{o}) = 0.5$
$\text{sin}(50^ ext{o}) \approx 0.7660$

Thus,

$R \approx 15 \cdot \frac{0.5}{0.7660} \approx 9.79 \, \text{N}$

Step 2

(b) the value of X

99%

104 rated

Answer

Next, to find the value of the force Q (magnitude X), we apply the cosine rule:

$R^2 = X^2 + 15^2 - 2 \cdot X \cdot 15 \cdot \text{cos}(100^ ext{o})$

Substituting the known values:

From (a), we have $R \approx 9.79$
$\text{cos}(100^ ext{o}) \approx -0.1736$

Thus,

$9.79^2 = X^2 + 15^2 - 2 \cdot X \cdot 15 \cdot (-0.1736)$

This simplifies to:

$96.04 = X^2 + 225 + 5.208 \cdot X$

Rearranging gives us a quadratic equation:

$X^2 + 5.208 X - 128.96 = 0$

Using the quadratic formula:

$X = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}$

where,

$a = 1$
$b = 5.208$
$c = -128.96$

Calculating the roots, we find:

$X \approx 19.3 \, \text{N}$

Two forces P and Q act on a particle at a point O - Edexcel - A-Level Maths Mechanics - Question 5 - 2008 - Paper 1

Worked Solution & Example Answer:Two forces P and Q act on a particle at a point O - Edexcel - A-Level Maths Mechanics - Question 5 - 2008 - Paper 1

(a) the magnitude of R

(b) the value of X

Join the A-Level students using SimpleStudy...