Lamina on surface with string or rod support

5 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-3_348_1205_251_470} \(A B C\) is a uniform triangular lamina of weight 19 N , with \(A B = 0.22 \mathrm {~m}\) and \(A C = B C = 0.61 \mathrm {~m}\). The plane of the lamina is vertical. \(A\) rests on a rough horizontal surface, and \(A B\) is vertical. The equilibrium of the lamina is maintained by a light elastic string of natural length 0.7 m which passes over a small smooth peg \(P\) and is attached to \(B\) and \(C\). The portion of the string attached to \(B\) is horizontal, and the portion of the string attached to \(C\) is vertical (see diagram).

1. Show that the tension in the string is 10 N .
2. Calculate the modulus of elasticity of the string.
3. Find the magnitude and direction of the force exerted by the surface on the lamina at \(A\).

1 A uniform semicircular lamina has diameter \(A B\) of length 0.8 m .

1. Find the distance of the centre of mass of the lamina from \(A B\). The lamina rests in a vertical plane, with the point \(B\) of the lamina in contact with a rough horizontal surface and with \(A\) vertically above \(B\). Equilibrium is maintained by a force of magnitude 6 N in the plane of the lamina, applied to the lamina at \(A\) and acting at an angle of \(20 ^ { \circ }\) below the horizontal.
2. Calculate the mass of the lamina.

1 A particle \(P\) is moving in a circle of radius 2 m . At time \(t\) seconds, its velocity is \(( t - 1 ) ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant.
[0pt] [5] \includegraphics[max width=\textwidth, alt={}, center]{0f39ff02-a4fc-49ce-b87e-f70bef5a58b6-04_591_805_262_671} A uniform square lamina \(A B C D\) of side \(4 a\) and weight \(W\) rests in a vertical plane with the edge \(A B\) inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(A B\), where \(B E = 3 a\) (see diagram).

1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\).
2. Given that the lamina is about to slip, find the value of \(\mu\).

1 A particle \(P\) is moving in a circle of radius 2 m . At time \(t\) seconds, its velocity is \(( t - 1 ) ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant.
[0pt] [5] \includegraphics[max width=\textwidth, alt={}, center]{4240c99e-10ba-443e-8021-1872e6e64ccf-04_591_805_262_671} A uniform square lamina \(A B C D\) of side \(4 a\) and weight \(W\) rests in a vertical plane with the edge \(A B\) inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(A B\), where \(B E = 3 a\) (see diagram).

1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\).
2. Given that the lamina is about to slip, find the value of \(\mu\).