<p>A submarine is a vessel that can be submerged and navigated under water, usually built for warfare and armed with torpedoes or guided missiles. A tugboat on the other hand, is a strongly built powerful boat used for towing and pushing. The example below illustrates how to find the resultant force exerted by tugboats on a submarine.</p>
<div class="ns-school-emp">
<h2>Example: Finding the Resultant in Two Dimensions Graphically</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>A number of tugboats are trying to manoeuvre a submarine in the harbour but they are not working as a team. Each tugboat is exerting a different force on the submarine.</p>
<img alt="" src="https://data.ulearngo.com/assets/0a624ae9-4d36-4644-ad9a-f4aec3efb0a6"/>
<p>Given the following force vectors, determine the resultant force:</p>
<ul>
<li>\(\vec{F}_{1}\) = \(\text{3.4}\) \(\text{kN}\) in the positive \(x\)-direction</li>
<li>\(\vec{F}_{2}\) = \(\text{4 000}\) \(\text{N}\) in the positive \(y\)-direction</li>
<li>\(\vec{F}_{3}\) = \(\text{300}\) \(\text{N}\) in the negative \(y\)-direction</li>
<li>\(\vec{F}_{4}\) = \(\text{7}\) \(\text{kN}\) in the negative \(y\)-direction</li>
</ul>
</div>
<div>
<h3>Step 1: Convert to consistent S.I. units</h3>
<p>To use the graphical method of finding the resultant we need to work in the same units. Strictly speaking in this problem all the vectors are in newtons but they have different factors which will affect the choice of scale. These need to taken into account and the simplest approach is to convert them all to a consistent unit and factor. We could use kN or N, the choice does not matter. We will choose kN. Remember that k represents a factor of \(\times 10^{3}\).</p>
<p>\(\vec{F}_{1}\) and \(\vec{F}_{4}\) do not require any adjustment because they are both in kN. To convert N to kN we use: \begin{align*} \text{kN} &amp;= \times 10^{3}\\ \frac{\text{N}}{\text{kN}} &amp;= \frac{1}{\times 10^{3}}\\ \text{N} &amp;= \times 10^{-3}\ \text{kN} \end{align*}</p>
<p>To convert the magnitude of \(\vec{F}_{2}\) to kN: \begin{align*} F_2 &amp;= \text{4 000}\text{ N}\\ F_2 &amp;= \text{4 000} \times \text{10}^{-\text{3}}\text{ kN}\\ F_2 &amp;= \text{4}\text{ kN} \end{align*} Therefore \(\vec{F}_{2}\) = \(\text{4}\) \(\text{kN}\) in the positive \(y\)-direction.</p>
<p>To convert the magnitude of \(\vec{F}_{3}\) to kN: \begin{align*} F_3 &amp;= \text{300}\text{ N}\\ F_3 &amp;= \text{300} \times \text{10}^{-\text{3}}\text{ kN}\\ F_3 &amp;= \text{0,3}\text{ kN} \end{align*} Therefore \(\vec{F}_{3}\) = \(\text{0.3}\) \(\text{kN}\) in the negative \(y\)-direction. So:</p>
<ul>
<li>\(\vec{F}_{1}\) = \(\text{3.4}\) \(\text{kN}\) in the positive \(x\)-direction</li>
<li>\(\vec{F}_{2}\) = \(\text{4}\) \(\text{kN}\) in the positive \(y\)-direction</li>
<li>\(\vec{F}_{3}\) = \(\text{0.3}\) \(\text{kN}\) in the negative \(y\)-direction</li>
<li>\(\vec{F}_{4}\) = \(\text{7}\) \(\text{kN}\) in the negative \(y\)-direction</li>
</ul>
</div>
<div>
<h3>Step 2: Choose a scale and draw axes</h3>
<p>The vectors we have do have very big magnitudes so we need to choose a scale that will allow us to draw them in a reasonable space, we can use \(\text{1}\) \(\text{kN}\) : \(\text{1}\) \(\text{cm}\) as our scale for the drawings.</p>
</div>
<div>
<h3>Step : Determine \(\vec{R}_{x}\)</h3>
<p>There is only one vector in the \(x\)-direction, \(\vec{F}_{1}\), therefore \(\vec{R}_{x}\) = \(\vec{F}_{1}\).</p>
</div>
<div>
<h3>Step 3: Determine \(\vec{R}_{y}\)</h3>
<p>Then we determine the resultant of all the vectors that are parallel to the \(y\)-axis. There are three vectors \(\vec{F}_{2}\), \(\vec{F}_{3}\) and \(\vec{F}_{4}\) that we need to add. We do this using the tail-to-head method for co-linear vectors.</p>
<img alt="38a8d0502fc37c396a7548151944e5a5.png" src="https://data.ulearngo.com/assets/128a2d3a-556e-405e-9ba9-6e87b9dab695"/>
<p>The single vector, \(\vec{R}_{y}\), that would give us the same outcome is:</p>
<img alt="0b48bcccd364975296199bdc652fa237.png" src="https://data.ulearngo.com/assets/4672f62d-cb0d-49ee-9db3-da99d97a3469"/></div>
<div>
<h3>Step 4: Draw axes</h3>
<p>Then we draw axes that the diagram should fit on. We need our axes to extend just further than the vectors aligned with each axis. Our axes need to start at the origin and go beyond \(\text{3.4}\) \(\text{kN}\) in the positive \(x\)-direction and further than \(\text{3.3}\) \(\text{kN}\) in the negative \(y\)-direction. Our scale choice of \(\text{1}\) \(\text{kN}\) : \(\text{1}\) \(\text{cm}\) means that our axes actually need to extend \(\text{3.4}\) \(\text{cm}\) in the positive \(x\)-direction and further than \(\text{3.3}\) \(\text{cm}\) in the negative \(y\)-direction</p>
<img alt="3f9aada67904ba123a9689f819833bec.png" src="https://data.ulearngo.com/assets/bfe576f0-2eb8-4cb7-941e-03567bb45cb2"/></div>
<div>
<h3>Step 5: Draw \(\vec{R}_x\)</h3>
<p>The length of \(\vec{R}_x\) is \(\text{3.4}\) \(\text{kN}\) so the arrow we need to draw must be \(\text{3.4}\) \(\text{cm}\) long. The arrow must point in the positive \(x\)-direction.</p>
<img alt="70613b64aa40cbec157e0d1e6daa0598.png" src="https://data.ulearngo.com/assets/cf6ef571-23b5-4ddc-988c-4ff413468a86"/></div>
<div>
<h3>Step 6: Draw \(\vec{R}_y\)</h3>
<p>The length of \(\vec{R}_y\) is \(\text{3.3}\) \(\text{kN}\) so the arrow we need to draw must be \(\text{3.3}\) \(\text{cm}\) long. The arrow must point in the negative \(y\)-direction. The important fact to note is that we are implementing the head-to-tail method so the vector must start at the end (head) of \(\vec{R}_{x}\).</p>
<img alt="6bb5e6f7ad7af83b4fd9271327a067a6.png" src="https://data.ulearngo.com/assets/7d5f4800-cbb7-4ea5-bab9-157ae1c5cb02"/></div>
<div>
<h3>Step 7: Draw the resultant vector, \(\vec{R}\)</h3>
<p>The resultant vector is the vector from the tail of the first vector we drew directly to the head of the last vector we drew. This means we need to draw a vector from the tail of \(\vec{R}_{x}\) to the head of \(\vec{R}_{y}\).</p>
<img alt="bceb97b8ceeb5277d334a1776b2cb9e3.png" src="https://data.ulearngo.com/assets/fc2483bb-bf31-47de-a09a-d00da92c6b02"/></div>
<div>
<h3>Step 8: Measure the resultant, \(\vec{R}\)</h3>
<p>We are solving the problem graphically so we now need to measure the magnitude of the vector and use the scale we chose to convert our answer from the diagram to the actual result. In the last diagram the resultant, \(\vec{R}\) is \(\text{4.7}\) \(\text{cm}\) long therefore the magnitude of the vector is \(\text{4.7}\) \(\text{kN}\).</p>
<p>The direction of the resultant we need to measure from the diagram using a protractor. The angle that the vector makes with the \(x\)-axis is \(\text{44}\)\(\text{°}\).</p>
</div>
<div>
<h3>Step 9: Quote the final answer</h3>
<p>\(\vec{R}\) is \(\text{4.7}\) \(\text{kN}\) at \(-\text{44}\)\(\text{°}\) from the positive \(x\)-direction.</p>
</div>
</div>
</div>

1c4ddb66-ad36-4f81-90e7-4629f497d0b4

Resultant Force on a Submarine

Physics is a science discipline concerned with the nature and properties of matter and energy. Physics topics include mechanics, heat, light and radiation, sound, electricity, magnetism, the structure of atoms, and so on.

Physics

Discover the principles and operations of scalars and vectors such as graphical and algebraic techniques of vector addition, finding magnitude with Pythagoras theorem, and resultant force on a submarine.


Resultant Force on a Submarine

Track Your Learning Progress