Answer to Question #272769 in Classical Mechanics for joseph

Question

1. The cart is allowed to roll freely down the inclined plane due to its weight. Show that the slope of the surface of the liquid, theta. during the motion is theta = phi.

2. The cart is given a constant acceleration a up the inclined plane. Show that the lines of constant pressure within the liquid have a slope of tan theta = (a cos phi)/ (a sin phi + g).

3. A large container of benzene is transported on the truck. Determine the level in each of the vent tubes A and B if the truck accelerates at a = 1.5 m/s². When the truck is at rest, h_A = h_B = 0.4 m.

4. A woman stands on a horizontal platform that is rotating at a constant angular velocity w. If she is holding a cup of tea, and the center of the cup is 0.9 m from the axis of rotation, determine w if the slope angle of the liquid's surface is 5°. Neglect the size of the cup.

 

 

Answer 1

1.

$a=g\sin\phi,$

$a_x=-(g\sin \phi)\cos \phi,$

$a_y=-(g\sin \phi)\sin\phi,$

$\tan\theta=-\frac{a_x}{g+a_y}=\frac{g\sin \phi\cos\phi}{g-g\sin^2\phi}=\frac{\sin\phi\cos\phi}{\cos^2\phi}=\frac{\sin\phi}{\cos\phi}=\tan \phi,$

$\theta=\phi,$

2.

$\frac{d_y}{d_x}=-\frac{a_x}{g+a_y}=-\tan \theta,$

$\tan \theta=\frac{a_x}{g+a_y},$

$a_x=a\cos\phi,$

$a_y=a\sin\phi,$

$\tan\theta=\frac{a\cos\phi}{a\sin\phi+g},$

3.

$\tan\theta=\frac ag,$

$\Delta h=\frac{AB}2\tan \theta=0.23~m,$

$h_A'=h_A-\Delta h=0.17~m,$

$h_B'=h_B+\Delta h=0.63~m,$

4.

$\tan \theta=\frac ag=\frac{\omega^2 r}g,$

$\omega=\sqrt{\frac{g\tan \theta}r}=0.95~\frac{rad}s.$