# Just a small conceptual issue regarding the basics of vectors In the figure, there are two blocks of masses m and 2m connected by a light string passing over a frictionless pulley. The mass m is placed on a smooth inclined plane of inclination 30 degrees and 2m hangs vertically.

Now we gotta find the acceleration when the system is released.

Solution: acceleration=net force/total mass = 2mg-mgsin30/3m

Now I dont get why is it mgSin30. Like according to trigo sin30=mg/F where F is the force the block m is putting on the system. And since we have to put the net force in the answer. Shouldn't it be: force 2m is applying - force m block is applying/3m

so shouldnt it be F=mg/sin30 So i dont get why are we supposed to use mgsin30.

## 3 Answers

The values of sin tables use the length of the sides, not the force applied.

Taking components on the block of mass m, We find that Mg splits in two components, Mgsin30 parallel to the plane of the wedge and Mgcos30, perpendicular to the plane.

To understand the result you need draw a set of coordinate axes in which to decompose the "vector". What we are doing when we apply Newton's second law is equating vector components in the independent directions of space. In this problem, with no movement out of the page, we have 2 dimensions so every vector will have 2 components, one of which might be zero.

For this type of problem it is customary to decompose the forces acting on the "tilted" block in a coordinate system with one axis along the ramp and the other perpendicular to that ramp. Most books will tell you that this is "easier" because the normal force has only one component (in the direction perpendicular to the ramp), and if there is friction that will also have only one component (parallel to the ramp). The weight of the block needs to be decomposed in to two components, parallel and perpendicular to the ramp. Here is where you need to draw the triangles generated by your coordinates carefully and figure out which direction gets the sine and which the cosine. This is determined by the geometry of the wedge AND your choice of coordinates. I cannot stress this enough, you will get completely different components for different choices of coordinates. For this reason I do not think it's reasonable to cite "an answer". If you re-post the pic with a set of coordinates drawn any one of us can tell you what the components of mg are in that set of coordinates. The final answer should be the same no matter what coordinates you choose.

Also, you seem to be confusing something about the application of Newton's law to this problem as "the force on the block" is not what is being decomposed. It would help to go through the steps carefully.

The force on the left side mass which is opposing the motion is the component of gravity which is parallel to the surface. The given 30 degree angle is also found between the direction of gravity and the normal to the surface. This can be verified by imagining the slope of the surface dropping toward zero. To find the component of a vector which is adjacent to the angle, you multiply by the cosine. In this case we want the component which is opposite the angle. So you multiply by the sine. (By the way, with pulley systems, I like to wrap my x axis around the pulley. In this case with positive x down on the right side.)