Alright students, let's solve the problem step by step to determine the magnitude of the tension force when the box is pulled up a ramp.
Given:
- The net force in the [tex]\( x \)[/tex] direction is [tex]\( 98 \, \text{N} \)[/tex].
- The angle of the ramp with the horizontal surface is [tex]\( 22^\circ \)[/tex].
We're looking for the magnitude of the force of tension, denoted as [tex]\( F_{\text{tension}} \)[/tex].
1. Understanding the Components:
- The net force in the [tex]\( x \)[/tex] direction ([tex]\( F_{\text{net}, x} \)[/tex]) is given.
- The force of tension [tex]\( F_{\text{tension}} \)[/tex] has components in both the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] directions.
- The component of the tension force in the [tex]\( x \)[/tex] direction is [tex]\( F_{\text{tension}} \cos(\theta) \)[/tex], where [tex]\( \theta \)[/tex] is [tex]\( 22^\circ \)[/tex].
2. Setting up the Equation:
- From the net force in the [tex]\( x \)[/tex] direction, we get:
[tex]\[
F_{\text{net}, x} = F_{\text{tension}} \cos(22^\circ)
\][/tex]
3. Solving for [tex]\( F_{\text{tension}} \)[/tex]:
- Rearrange the equation to solve for [tex]\( F_{\text{tension}} \)[/tex]:
[tex]\[
F_{\text{tension}} = \frac{F_{\text{net}, x}}{\cos(22^\circ)}
\][/tex]
- Given [tex]\( F_{\text{net}, x} = 98 \, \text{N} \)[/tex]:
[tex]\[
F_{\text{tension}} = \frac{98 \, \text{N}}{\cos(22^\circ)}
\][/tex]
4. Calculation:
- We can use a calculator to find [tex]\( \cos(22^\circ) \)[/tex]:
[tex]\[
\cos(22^\circ) \approx 0.9272
\][/tex]
- Substituting this value:
[tex]\[
F_{\text{tension}} = \frac{98 \, \text{N}}{0.9272} \approx 105.696 \, \text{N}
\][/tex]
- Round this result to the nearest integer:
[tex]\[
F_{\text{tension}} \approx 106 \, \text{N}
\][/tex]
5. Conclusion:
- The magnitude of the force of tension needed to have a net force of [tex]\( 98 \, \text{N} \)[/tex] in the [tex]\( x \)[/tex]-direction on a ramp inclined at [tex]\( 22^\circ \)[/tex] is approximately [tex]\( 106 \, \text{N} \)[/tex].
Therefore, the correct answer among the given choices is not explicitly listed. However, considering the rounded value of [tex]\( 106 \, \text{N} \)[/tex], this value closely approximates the scenario given.
