Moments

A uniform beam $AB$ has length 6 m and mass 20 kg. It rests horizontally on two supports, one at $A$ and one at $B$. A particle $P$ of mass 15 kg is placed on the beam at a distance 2 m from $A$, as shown.

Find the magnitude of the reaction on the beam at each support. [4]

A uniform plank $AB$ has length 5 m and mass 30 kg. It rests horizontally on two supports at $C$ and $D$, where $AC=1$ m and $DB=1.5$ m, as shown. A man of mass 60 kg stands on the plank at its midpoint. The man is modelled as a particle.

(a) Find the magnitude of the reaction on the plank at $C$ and at $D$. [4]

The man now walks along the plank towards $B$.

(b) Find the distance of the man from $B$ when the plank is about to tilt. [3]

(c) State how you have used the modelling assumption that the plank is uniform. [1]

A uniform ladder $AB$, of length 4 m and mass 12 kg, rests with its end $A$ on rough horizontal ground and its end $B$ against a smooth vertical wall. The ladder makes an angle $\theta$ with the ground and lies in a vertical plane perpendicular to the wall. The coefficient of friction between the ladder and the ground is 0.4.

The ladder is in limiting equilibrium.

(a) Show that $\tan\theta=\frac54$. [5]

(b) Hence find the size of angle $\theta$. [1]

A non-uniform rod $AB$ has length 3 m and mass 8 kg. The rod hangs in equilibrium in a horizontal position, supported by two vertical light strings attached to the rod at $A$ and at $C$, where $AC=2.4$ m, as shown. The tension in the string at $C$ is three times the tension in the string at $A$.

(a) Find the distance of the centre of mass of the rod from $A$. [4]

A particle of mass $m$ kg is now attached to the rod at $A$. The rod remains horizontal and in equilibrium, and the tensions in the two strings are now equal.

(b) Find the value of $m$. [3]