What Is the Acceleration of Two Blocks on Inclined Planes?

[jhson114]

Two identical blocks tied together with a string which passes over a pulley at the crest of the inclined planes, one of which makes an angle q1 = 20° to the horizontal, the other makes the complementary angle q2 = 70°. If there is no friction anywhere, with what acceleration do the blocks move?
heres a simple pic to describe the question better:
https://tychosrv-s.phys.washington.edu/cgi/courses/shell/common/showme.pl?courses/phys121/autumn04/homework/04/two_blocks_on_incline1_NWT/5.gif

I've been working on this for like an hour drawing fbd and everything but can't seem to find the answer. help please

 

[jhson114]

after much after. i figured it out. :)

 

[wais]

Based on the given information and the free body diagrams, we can use Newton's Second Law to determine the acceleration of the blocks. Since there is no friction, the only forces acting on the blocks are the tension in the string and the weight of each block.

Let's start with the block on the 20° incline. The weight of this block can be resolved into two components, one parallel to the incline and one perpendicular to it. The component parallel to the incline will contribute to the acceleration of the block, while the perpendicular component will cancel out with the normal force from the incline.

Using trigonometry, we can find that the parallel component of the weight is mg*sin(20°). This is equal to the tension in the string, so we can set up the following equation:

mg*sin(20°) = T

Next, let's look at the block on the 70° incline. Similar to the first block, we can find that the parallel component of the weight is mg*sin(70°). This will also be equal to the tension in the string, so we have:

mg*sin(70°) = T

Since both blocks are tied together, the tension in the string is the same for both blocks. We can set these two equations equal to each other and solve for the acceleration:

mg*sin(20°) = mg*sin(70°)
sin(20°) = sin(70°)
0.342 = 0.939

This is clearly not true, so there must be something wrong with our assumptions. Upon further inspection, we can see that the blocks will not move in this situation. The tension in the string is balancing out the components of the weight, resulting in a net force of zero and therefore no acceleration.

In order for the blocks to move, there needs to be a difference in the tension in the string. This could be achieved by adding friction to one of the inclines or by changing the masses of the blocks. Without any external forces, the blocks will not move and will remain in equilibrium.