A uniform rod, [AB], of length 4 m and weight 80 N is smoothly hinged at end A to a horizontal floor. One end of a light inelastic string is attached to B and the other end of the string is attached to a horizontal ceiling. The string makes an angle of 60° with the ceiling and the rod makes an angle of 30° with the floor, as shown in the diagram. The rod is in equilibrium. (i) Show on a diagram all the forces acting on the rod [AB]. (ii) Write down the two equations that arise from resolving the forces horizontally and vertically. (iii) Write down the equation that arises from taking moments about the point A. (iv) Find the tension in the string. (v) Find the magnitude of the reaction at the hinge, A.

Leaving Cert Applied Maths Statics: A uniform rod, [AB], of length 4

A uniform rod, [AB], of length 4 m and weight 80 N is smoothly hinged at end A to a horizontal floor - Leaving Cert Applied Maths - Question 7 - 2012

Question 7

A uniform rod, [AB], of length 4 m and weight 80 N is smoothly hinged at end A to a horizontal floor. One end of a light inelastic string is attached to B and the o... show full transcript

Worked Solution & Example Answer:A uniform rod, [AB], of length 4 m and weight 80 N is smoothly hinged at end A to a horizontal floor - Leaving Cert Applied Maths - Question 7 - 2012

Step 1

Show on a diagram all the forces acting on the rod [AB].

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Answer

The forces acting on the rod [AB] are:

The weight of the rod (80 N) acting downwards at its center of gravity (which is at the midpoint of the rod).
The tension in the string (T), acting at point B, directed along the string at an angle of 60° to the ceiling.
The reaction at point A, which can be broken into horizontal (X) and vertical (Y) components.

Step 2

Write down the two equations that arise from resolving the forces horizontally and vertically.

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Answer

Resolving the forces horizontally:

$X = T imes ext{cos}(60°)$

Resolving the forces vertically:

$Y + T imes ext{sin}(60°) = 80$

Step 3

Write down the equation that arises from taking moments about the point A.

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Answer

Taking moments about point A gives:

$T imes 4 = 80 imes 2 imes ext{cos}(30°)$

Step 4

Find the tension in the string.

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Answer

From the moments equation:

$T imes 4 = 80 imes 2 imes ext{cos}(30°)$

Solving for T:

$T = rac{80 imes 2 imes ext{cos}(30°)}{4} = 20 imes rac{ ext{sqr}(3)}{2} = 20 ext{/3}$

Step 5

Find the magnitude of the reaction at the hinge, A.

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Answer

To find the reaction at hinge A, we can use:

$R = ext{sqr}igg{(}igg{(}T imes ext{cos}(60°)igg{)}^2 + igg{(}Y - 80 + T imes ext{sin}(60°)igg{)}^2igg{)}$

Substituting the found values yields:

$R = rac{ ext{sqr}igg{(}(0 + 50igg{)}^2 + (80 - T imes ext{sin}(60°))^2}{20/ ext{sqr}(3)}}$

A uniform rod, [AB], of length 4 m and weight 80 N is smoothly hinged at end A to a horizontal floor - Leaving Cert Applied Maths - Question 7 - 2012

Worked Solution & Example Answer:A uniform rod, [AB], of length 4 m and weight 80 N is smoothly hinged at end A to a horizontal floor - Leaving Cert Applied Maths - Question 7 - 2012

Show on a diagram all the forces acting on the rod [AB].

Write down the two equations that arise from resolving the forces horizontally and vertically.

Write down the equation that arises from taking moments about the point A.

Find the tension in the string.

Find the magnitude of the reaction at the hinge, A.

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