A sign outside a pub is attached to a light rod of length 1 m which is freely hinged to the wall and supported in a vertical plane by a light string as in the diagram. The sign is assumed to be a uniform rectangle of mass 10 kg. The angle of the string to the horizontal is 25°.

i) Find the tension in the string.
ii) Find the magnitude and direction of the reaction force of the hinge on the sign.

Question

A sign outside a pub is attached to a light rod of length 1 m which is freely hinged to the wall and supported in a vertical plane by a light string as in the diagram. The sign is assumed to be a uniform rectangle of mass 10 kg. The angle of the string to the horizontal is 25°.

i) Find the tension in the string.
ii) Find the magnitude and direction of the reaction force of the hinge on the sign.

Accepted Answer

i) The diagram shows the forces acting on the rod, where [latex]R_{H}[/latex] and [latex]R_{V}[/latex] are the magnitudes of the horizontal and vertical components of the reaction R on the rod at the wall.

Taking moments about O

R × 0 - 10g × 0.5 + T sin 25° × 1 = 0

⇒ T sin 25° = 5g

T = 116

The tension is 116 N.

ii) You can resolve to find the reaction at the wall.

Horizontally: [latex]R_{H} [/latex] = T cos 25°

⇒ [latex]R_{H} [/latex] = 105

Vertically: [latex]R_{V}[/latex] + T sin 25° = 10g

⇒ [latex]R_{V}[/latex] = 10g - 5g = 49

R = [latex]\sqrt{105² + 49²}[/latex]

= 116

θ = arctan [latex](\frac{49}{105})[/latex] = 25°

The reaction at the hinge has magnitude 116 N and acts at 25 above the horizontal.

Question 7.4: A sign outside a pub is attached to a light rod of length 1 ...

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