A uniform rod AB has length 3 m and weight 120 N. The rod rests in equilibrium in a horizontal position, smoothly supported at points C and D, where AC = 0.5 m and AD = 2 m, as shown in Fig. 3. A particle of weight W newtons is attached to the rod at a point E where AE = x metres. The rod remains in equilibrium and the magnitude of the reaction at C is now twice the magnitude of the reaction at D. (a) Show that $W = \frac{60}{1 - x}$. (b) Hence deduce the range of possible values of x.

A-Level Edexcel Maths Mechanics Moments: A uniform rod AB has length 3 m

A uniform rod AB has length 3 m and weight 120 N - Edexcel - A-Level Maths Mechanics - Question 6 - 2003 - Paper 1

Question 6

A uniform rod AB has length 3 m and weight 120 N. The rod rests in equilibrium in a horizontal position, smoothly supported at points C and D, where AC = 0.5 m and A... show full transcript

Worked Solution & Example Answer:A uniform rod AB has length 3 m and weight 120 N - Edexcel - A-Level Maths Mechanics - Question 6 - 2003 - Paper 1

Step 1

Show that $W = \frac{60}{1 - x}$

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Answer

To establish the relationship between $W$ , the weight of the particle, and $x$ , we can apply the principle of moments around point D:

Setting Up the Equation: The moments about point D should balance since the rod is in equilibrium. Thus,

$W \times (2 - x) + 120 \times 1.5 = R \times 3$

Here, $R$ represents the reaction force at point C.
Expressing R: From the equilibrium of forces vertically:

$R = 3W + 120$
Substituting into Moment Equation: Substitute $R$ into the moment equation:

$W(2 - x) + 120 \times 1.5 = (3W + 120) \times 3$
Expanding and Simplifying: This leads to:

$W(2 - x) + 180 = 9W + 360$

Rearranging gives:

$W(2 - x) - 9W + 180 - 360 = 0$

Which simplifies to:

$W(1 - x) = 60$
Final Results: Thus,

$W = \frac{60}{1 - x}$ .

Step 2

Hence deduce the range of possible values of x

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Answer

From the equation we derived:

$W = \frac{60}{1 - x}$ ,

We can identify the conditions for $W$ to be valid:

Condition for W: For $W$ to be positive, we need:

$1 - x > 0 \implies x < 1$ .
Minimum Value of W: Also, $W$ must be greater than zero, requiring:

$1 - x > 0\implies x < 1$ ,

and we cannot have $x$ reaching or exceeding 1.

Thus, the only condition left is ensuring that $W$ is positive:

$W > 0 \implies 0 < x < 1$ .

Conclusion: Hence, the range of possible values of $x$ is:

$0 < x < 1$ .

A uniform rod AB has length 3 m and weight 120 N - Edexcel - A-Level Maths Mechanics - Question 6 - 2003 - Paper 1

Worked Solution & Example Answer:A uniform rod AB has length 3 m and weight 120 N - Edexcel - A-Level Maths Mechanics - Question 6 - 2003 - Paper 1

Show that $W = \frac{60}{1 - x}$

Hence deduce the range of possible values of x

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