Question

A cart is pushed up a ramp with a force of 250 N. The ramp sits at a 28° angle with the horizontal.A, Determine the components of the vector representing the force.B. Describe what would happen if the angle of the ramp was decreased?

Answer 1

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:

Component x of the force

We can find this component as follows:

`\begin{gathered} cos(28^(\circ))=(x)/(250N) \\ \\ cos(28^(\circ))250N=x \\ \\ x=cos(28^(\circ))250N=220.736898215N \\ \\ x=220.736898215N \end{gathered}`

Component y of the force

In this case, instead of using cosine, we have to use the sine function to find the y component of the force:

`\begin{gathered} sin(28^(\circ))=(y)/(250N) \\ \\ 250Nsin(28^(\circ))=y \\ \\ y=250Nsin(28^(\circ))=117.367890696N \\ \\ y=117.367890696N \\ \end{gathered}`

Therefore, the components of the vector representing the force are as follows:

`\langle220.736898215,117.367890696\rangle`

If we round the results to the nearest hundredth, we can rewrite the components as follows:

`\langle220.74,117.37\rangle`

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:

`\begin{gathered} x=cos(15^(\circ))250N=241.481456572N \\ \\ y=sin(15^(\circ))250N=64.7047612756N \end{gathered}`

Therefore, in summary, we have:

Part A: The components of the vector representing the force are:

`\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}`

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.