Questions: Problem 8: (11% of Assignment Value) A uniform beam of length L=2.7 m and mass M=39 kg has its lower end fixed to pivot at a point P on the floor, making an angle θ=17° as shown in the diagram. A horizontal cable is attached at its upper end B to a point A on a wall. A box of the same mass M as the beam is suspended from a rope that is attached to the beam one-fourth L from its upper end. Part (a) What is the y-component Py of the force, in newtons, exerted by the pivot on the beam? Py=

Problem 8: (11% of Assignment Value) A uniform beam of length L=2.7 m and mass M=39 kg has its lower end fixed to pivot at a point P on the floor, making an angle θ=17° as shown in the diagram. A horizontal cable is attached at its upper end B to a point A on a wall. A box of the same mass M as the beam is suspended from a rope that is attached to the beam one-fourth L from its upper end. Part (a) What is the y-component Py of the force, in newtons, exerted by the pivot on the beam? Py=
Transcript text: Problem 8: ( $11 \%$ of Assignment Value) A uniform beam of length $L=2.7 \mathrm{~m}$ and mass $M=39 \mathrm{~kg}$ has its lower end fixed to pivot at a point P on the floor, making an angle $\theta=17^{\circ}$ as shown in the diagram. A horizontal cable is attached at its upper end B to a point A on a wall. A box of the same mass $M$ as the beam is suspended from a rope that is attached to the beam one-fourth $L$ from its upper end. Part (a) What is the $y$-component $P_{\mathrm{y}}$ of the force, in newtons, exerted by the pivot on the beam? \[ P_{\mathrm{y}}= \]
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Solution

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Solution Steps

Step 1: Analyze the forces acting on the beam

The forces acting on the beam are:

  • Weight of the beam (Mg): Acts downwards at the center of the beam (L/2 from the pivot).
  • Weight of the box (Mg): Acts downwards at a distance of (3L/4) from the pivot.
  • Tension in the cable (T): Acts horizontally towards the wall at the upper end of the beam (L from the pivot).
  • Vertical component of the pivot force (Py): Acts upwards at the pivot.
  • Horizontal component of the pivot force (Px): Acts horizontally at the pivot.

We are interested in finding Py.

Step 2: Apply the equilibrium condition for vertical forces

Since the beam is in equilibrium, the sum of the vertical forces must be zero. $P_y - Mg - Mg = 0$ $P_y = 2Mg$

Step 3: Substitute the given values

$M = 39 kg$ $g = 9.8 m/s^2$ $P_y = 2 * 39 kg * 9.8 m/s^2$

Final Answer:

$P_y = 764.4 N$

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