We have a cart that is pushed up a ramp, and we have that the force is 250 N. We also know that the ramp sits at 28 degrees with the horizontal. Then we can draw the situation as follows:
Therefore, we can find the components of the force by finding the measures of x and y using trigonometric ratios (sine and cosine) as follows:
We can find this component as follows:
[tex]\begin{gathered} cos(28^{\circ})=\frac{x}{250N} \\ \\ cos(28^{\circ})250N=x \\ \\ x=cos(28^{\circ})250N=220.736898215N \\ \\ x=220.736898215N \end{gathered}[/tex]In this case, instead of using cosine, we have to use the sine function to find the y component of the force:
[tex]\begin{gathered} sin(28^{\circ})=\frac{y}{250N} \\ \\ 250Nsin(28^{\circ})=y \\ \\ y=250Nsin(28^{\circ})=117.367890696N \\ \\ y=117.367890696N \\ \end{gathered}[/tex]Therefore, the components of the vector representing the force are as follows:
[tex]\langle220.736898215,117.367890696\rangle[/tex]If we round the results to the nearest hundredth, we can rewrite the components as follows:
[tex]\langle220.74,117.37\rangle[/tex]What would happen if the angle of the ramp was decreased?
We can use a smaller angle to see what would happen in this case. We can determine what would happen if the angle of the ramp were 15 degrees with the horizontal. We have a similar representation as before:
We can see that the component y would decrease, and the x component will increase - we need to remember that the sum of both components needs to be equal to 250N, and we can check this as follows:
[tex]\begin{gathered} x=cos(15^{\circ})250N=241.481456572N \\ \\ y=sin(15^{\circ})250N=64.7047612756N \end{gathered}[/tex]Therefore, in summary, we have:
Part A: The components of the vector representing the force are:
[tex]\begin{gathered} \langle220.736898215,117.367890696\rangle \\ \\ \text{ If we round this result to the nearest hundredth, we have:} \\ \\ \langle220.74,117.37\rangle \\ \end{gathered}[/tex]Part B: We have checked that if the angle of the ramp was decreased, then the component x of the vector would increase (in measure), and the component y of the vector would decrease (in measure) - we need to remember that the sum of both components needs to be equal to 250N.
