Rob is pushing a block of 12 kg up a frictionless ramp. The ramp makes an angle of 35° with the horizontal. Assuming that Rob exerts a force of 148 N, find:
Let's represent what is happening in the problem with a simple sketch.
We need to represent a ramp that makes an angle of 35° with the horizontal, a block on it, and indicate that the block is pushed up by a force parallel to the ramp:
By looking at the sketch, we can conclude that there are three forces acting on the block:
Having figured out the forces acting on the block, let's draw a free-body diagram of the block:
Let's think about what we know, and what we're asked to find:
We know the mass of the block (12 kg), the angle the ramp makes with the horizontal (35°), and the force exerted by Rob (148 N).
We're asked to find the resultant force acting on the block, the acceleration that the block has as a result, and the normal force exerted by the ramp on the block.
Let's start by finding the resultant force.
In this case we have a block moving up a ramp, so for our convenience, we will use tilted coordinate axes, with the x axis in the direction of motion (uphill).
After drawing the coordinate axes on the free-body diagram of the block, we proceed to find the components of the individual forces acting on the block:
Keep in mind that the angle that the gravitational force, mg, makes with its y component, mgy, is equal to the angle that the ramp makes with the horizontal (35° in this case).
And the components of the resultant force will be:
We know that the motion of the block is along the ramp, therefore Ry must be zero. Otherwise there would be an acceleration in the y direction, which is not the case.
Therefore, the magnitude of R is equal to the absolute value of Rx. We know that Rx must be positive because the block is accelerating in the positive x direction. So, we can write:
We have already determined Rx in Eq. (1):
So, the resultant force on the block is 81 N and directed in the positive x direction:
Next we need to find the acceleration.
Since we know the resultant force and the mass, we can apply Newton's 2nd Law to get the acceleration:
|a =||81 N|
Lastly, we need to find the magnitude of the normal force.
The normal force N appears in Eq. (2):
And we've shown that Ry is zero, therefore:
At this point, we've found everything we were asked to find:
A heavy box of 135 kg is pushed up an inclined plane which makes an angle of 14.0° with the horizontal.
If the push has magnitude 550 N and the plane can be considered frictionless, what are: the resultant force acting on the box, the acceleration of the box, and the normal force?