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 Forces: 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 determine the value of W, we start by analyzing the equilibrium of the rod. We use the moments about point A:

Setting the clockwise moments equal to the counterclockwise moments:

The total moment about point A using the distances:
- The weight of the rod acts at its center, which is at 1.5 m.
- The reaction at point C (let's denote it as R) acts at the distance of 2 m from A.

The moment equation can be expressed as:

$W \cdot x + 120 \cdot 1.5 = R \cdot 2 + 2R \cdot 1$ Here, R is the reaction force at point C.

So, we have:

$R = \frac{W \cdot x + 180}{3}$

Now substituting R back into our moment equation:

$W \cdot x + 180 - 3R = W = 120$

From here we simplify to:

$W(1 - x) = 60$

Thus, we conclude:

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

Step 2

Hence deduce the range of possible values of x

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Answer

From the derived formula $W = \frac{60}{1 - x}$ , we analyze the constraints for W.

Since W represents the weight, it must be greater than 0: $W > 0 \rightarrow \frac{60}{1 - x} > 0$
This implies $1 - x > 0 \rightarrow x < 1$
Next, we consider the case where W must also remain finite, starting from 0:
- As W approaches infinite, it means 1 - x approaches 0, hence: $W > 0 \rightarrow x > 0$

Combining these inequalities, we conclude that:

$0 < x < 1$

Consequently, the possible range of values for x is (0, 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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