3.3.A

Video

Simulation

  • Two rigid bodies 1 and 2 are connected by a hinge at B.

  • Particle A on rigid body 1 is moving vertically with velocity \(v_A>0\).

  • Particle C on rigid body 2 is moving vertically.

  • Particle D on rigid body 2 is moving horizontally.

../../../_images/3.3.A.png

Given symbols: \(h\). Proceed as follows.

Steps

1. Velocities

Use arrows to define:

  • Velocity \(v_A\) of a particle at A (downward = positive).

  • Velocity \(v_C\) of a particle at C (downward = positive).

  • Velocity \(v_D\) of a particle at D (to the left = positive).

Solution

../../../_images/3.3.A_1.png

2. Angular Velocity of Body 2

Define the angular velocity \(\omega_2\) of rigid body 2 (clockwise = positive). Draw the respective arrow at the instant centre of rotation of body 2.

Solution

../../../_images/3.3.A_2.png

3. Angular Velocity of Body 1

Define the angular velocity \(\omega_1\) of rigid body 1 (counter clockwise = positive). Draw the respective arrow at the instant centre of rotation of body 1.

Solution

../../../_images/3.3.A_3.png

4. Comparing Angular Velocities

Show that:

\[\omega_1 = \omega_2\]

Solution

../../../_images/3.3.A_3.png
\[\begin{split}v_B &= r \omega_1 \\ &= r \omega_2\end{split}\]

So that:

\[\begin{split}r \omega_1 &= r \omega_2 \\\ \omega_1 &= \omega_2\end{split}\]

5. Velocity of A

Show that:

\[v_A = 3 v_D\]

Solution

../../../_images/3.3.A_3.png

Since \(\omega = \omega_1=\omega_2\):

\[\begin{split}v_A&=3 h \omega \\ v_D&=h \omega\end{split}\]

Therefore:

\[v_A=3 v_D\]

Find \(v_A\) for the following given quantities:

\[\begin{split}v_D &= 2.5\,\tfrac{\mathrm m}{\mathrm s} \\ h &= 10\,\mathrm m\end{split}\]

Solution

\[\begin{split}v_A &=3 v_D \\ &= 7.5\,\tfrac{\mathrm m}{\mathrm s}\end{split}\]

6. Angular Velocity of Body 2

Find the angular velocity of body 2 \(\omega_2\) in \(\tfrac{1}{\mathrm s}\) (1 per second) for the same following given quantities:

Solution

\[\begin{split}\omega_2 &= \tfrac{v_A}{3 h}\\ &= \tfrac{7.5\,\tfrac{\mathrm m}{\mathrm s}}{30\,\mathrm m} \\ &= 0.25 \,\tfrac{1}{\mathrm s}\end{split}\]