A letter L is made from a sheet of uniform thin plastic, with dimensions as shown in the diagram - Leaving Cert Applied Maths - Question 7 - 2014

To determine the angle \( \theta \) that line AD makes with the vertical, we can use trigonometric relationships.

From the center of mass calculated above, we have:

• Horizontal distance from D to CM (x) = 12.75 cm
• Vertical distance from D to CM (y) = 20.4 cm

Using the tangent function: $\tan \theta = \frac{x}{y} = \frac{12.75}{20.4}$

Calculating: ( \theta = \tan^{-1}\left( \frac{12.75}{20.4} \right) \)

This gives $\theta = 26.79°$.

For the rod diagram:

• Indicate forces including weight (85 N downwards), hinge reaction R, tension forces at P, and friction, ensuring they sum to zero in the x and y directions.

For the hemisphere diagram:

• Include forces such as weight (44 N downwards) and normal reaction from the floor, along with friction forces.

Using the equilibrium of forces:

• The normal reaction from the hemisphere is calculated as: $R = 85\cos(45^{\circ}) + 44 = 102.18 N$

Next, the frictional force can be given by: $F_{friction} = \mu R$

Where \( \mu \) is the coefficient of friction. By rearranging: $\mu = \frac{F_{friction}}{R}$.

To find the reaction at P (R):

• For vertical forces: \( ext{Total vertical:} \ Y = R\sin(45^{\circ}) = 72.25 N)
• For horizontal forces: \( ext{Total horizontal:} \ X = 85 - R\cos(45^{\circ}) = 12.75 N)
• Calculating R using Pythagorean theorem: $R = \sqrt{X^2 + Y^2} = \sqrt{(12.75)^2 + (72.25)^2} = 73.4 N$.