4. Exercises

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<div class="ovning">
<div class="ovning">
a) Find the magnitude of the force
a) Find the magnitude of the force
-
6 <math>\mathbf{i}</math><math>+</math> 8
+
6 <math>\mathbf{i}</math><math>+</math>8
<math>\mathbf{j}</math> N, where <math>\mathbf{i}</math> and <math>\mathbf{j}</math> are
<math>\mathbf{j}</math> N, where <math>\mathbf{i}</math> and <math>\mathbf{j}</math> are
perpendicular unit vectors.
perpendicular unit vectors.
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===Exercise 4.4===
===Exercise 4.4===
<div class="ovning">
<div class="ovning">
-
A force is expressed as 9 <math>\mathbf{i}</math><math>+</math> 10
+
A force is expressed as 9 <math>\mathbf{i}</math><math>-</math>10
<math>\mathbf{j}</math> N.
<math>\mathbf{j}</math> N.
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a)
a)
-
<math>\left( 4\mathbf{i}+3\mathbf{j} \right)\text{ N}</math>
+
(<math>4\mathbf{i}+3\mathbf{j}</math>
 +
) N
b)
b)
-
<math>\left( 4\mathbf{i}-3\mathbf{j} \right)\text{ N}</math>
+
(<math>4\mathbf{i}-3\mathbf{j}</math>
 +
) N
c)
c)
-
<math>\left( -2\mathbf{i}+8\mathbf{j} \right)\text{ N}</math>
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(<math>-2\mathbf{i}+8\mathbf{j}</math>
 +
) N
d)
d)
-
<math>\left( -5\mathbf{i}-7\mathbf{j} \right)\text{ N}</math>
+
(<math>-5\mathbf{i}-7\mathbf{j}</math>
 +
) N
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<div class="ovning">
<div class="ovning">
-
The diagram shows two forces and the unit vectors i and j.
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The diagram shows two forces and the unit vectors <math>\mathbf{i}</math> and <math>\mathbf{j}</math>.
-
 
+
 +
[[Image:E4.6.GIF]]
a) Express each force in the form <math>a\mathbf{i}+b\mathbf{j}</math>.
a) Express each force in the form <math>a\mathbf{i}+b\mathbf{j}</math>.
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</div>{{#NAVCONTENT:Answer a|Answer 4.6a|Answer b|Answer 4.6b|Answer c|Answer 4.6c|Solution a|Solution 4.6a|Solution b|Solution 4.6b|Solution c|Solution 4.6c}}
</div>{{#NAVCONTENT:Answer a|Answer 4.6a|Answer b|Answer 4.6b|Answer c|Answer 4.6c|Solution a|Solution 4.6a|Solution b|Solution 4.6b|Solution c|Solution 4.6c}}
 +
 +
 +
 +
 +
===Exercise 4.7===
 +
<div class="ovning">
 +
The diagram shows 3 forces and the perpendicular unit vectors <math>\mathbf{i}</math> and <math>\mathbf{j}</math>.
 +
 +
[[Image:E4.7.GIF]]
 +
 +
a) Express each force in terms of the unit vectors <math>\mathbf{i}</math> and <math>\mathbf{j}</math>.
 +
 +
b) Find the magnitude of the resultant of the 3 forces.
 +
 +
 +
</div>{{#NAVCONTENT:Answer|Answer 4.7|Solution a|Solution 4.7a|Solution b|Solution 4.7b}}
 +
 +
 +
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===Exercise 4.8===
 +
<div class="ovning">
 +
 +
A force
 +
<math>{\mathbf{F}_{1}}</math>
 +
= (<math>9\mathbf{i}+12\mathbf{j}</math>) N and a second force <math>{\mathbf{F}_{2}}</math> = (<math>-4\mathbf{i}+\mathbf{j}</math>) N act on a particle.
 +
 +
(a) Find the magnitude of
 +
<math>{\mathbf{F}_{1}}</math>.
 +
 +
 +
(b) Draw a diagram to show the direction of
 +
<math>{\mathbf{F}_{1}}</math>, and calculate the
 +
angle between
 +
<math>{\mathbf{F}_{1}}</math>
 +
and <math>\mathbf{i}</math>.
 +
 +
 +
(c) Find the resultant of
 +
<math>{\mathbf{F}_{1}}</math>
 +
and <math>{\mathbf{F}_{2}}</math>.
 +
 +
 +
</div>{{#NAVCONTENT:Answer a|Answer 4.8a|Answer b|Answer 4.8b|Answer c|Answer 4.8c|Solution a|Solution 4.8a|Solution b|Solution 4.8b|Solution c|Solution 4.8c}}
 +
 +
 +
===Exercise 4.9===
 +
<div class="ovning">
 +
Four forces are shown on the diagram below.
 +
 +
[[Image:E4.9.GIF]]
 +
 +
a) Find the magnitude of the resultant of the four forces.
 +
 +
b) Find the angle between the resultant force and the unit vector <math>\mathbf{i}</math>
 +
correct to the nearest degree.
 +
 +
 +
 +
</div>{{#NAVCONTENT:Answer|Answer 4.9|Solution a|Solution 4.9a|Solution b|Solution 4.9b}}

Current revision

       Theory          Exercises      


Exercise 4.1

The diagram shows a force and the unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j}

Image:E4.1.GIF

Express the force in terms of the unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j}

Answer
Answer 4.1
Solution
Solution 4.1


Exercise 4.2

Express each of the forces given below in the form \displaystyle a\mathbf{i}+b\mathbf{j}

Image:E4.2.GIF

Answer a
Answer 4.2a
Answer b
Answer 4.2b
Answer c
Answer 4.2c
Answer d
Answer 4.2d
Answer e
Answer 4.2e
Answer f
Answer 4.2f


Exercise 4.3

a) Find the magnitude of the force 6 \displaystyle \mathbf{i}\displaystyle +8 \displaystyle \mathbf{j} N, where \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are perpendicular unit vectors.

b) Find the angle, to the nearest degree, between the force and the unit vector \displaystyle \mathbf{i}.


Answer
Answer 4.3
Solution a
Solution 4.3a
Solution b
Solution 4.3b


Exercise 4.4

A force is expressed as 9 \displaystyle \mathbf{i}\displaystyle -10 \displaystyle \mathbf{j} N.

a) Calculate the magnitude of this force.

b) Find the angle between the force and the unit vector \displaystyle \mathbf{i}.

Answer
Answer 4.4
Solution a
Solution 4.4a
Solution b
Solution 4.4b

Exercise 4.5

For each force given below find its magnitude and the angle between the force and the unit vector \displaystyle \mathbf{i}.

a) (\displaystyle 4\mathbf{i}+3\mathbf{j} ) N

b) (\displaystyle 4\mathbf{i}-3\mathbf{j} ) N

c) (\displaystyle -2\mathbf{i}+8\mathbf{j} ) N

d) (\displaystyle -5\mathbf{i}-7\mathbf{j} ) N


Answer a
Answer 4.5a
Answer b
Answer 4.5b
Answer c
Answer 4.5c
Answer d
Answer 4.5d
Solution a
Solution 4.5a
Solution b
Solution 4.5b
Solution c
Solution 4.5c
Solution d
Solution 4.5d


Exercise 4.6

The diagram shows two forces and the unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j}.

Image:E4.6.GIF

a) Express each force in the form \displaystyle a\mathbf{i}+b\mathbf{j}.

b) Find the magnitude of the resultant of these two forces.

c) Find the angle between the resultant force and the unit vector \displaystyle \mathbf{i}, and draw a diagram to show the direction of the resultant force.

Answer a
Answer 4.6a
Answer b
Answer 4.6b
Answer c
Answer 4.6c
Solution a
Solution 4.6a
Solution b
Solution 4.6b
Solution c
Solution 4.6c



Exercise 4.7

The diagram shows 3 forces and the perpendicular unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j}.

Image:E4.7.GIF

a) Express each force in terms of the unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j}.

b) Find the magnitude of the resultant of the 3 forces.


Answer
Answer 4.7
Solution a
Solution 4.7a
Solution b
Solution 4.7b


Exercise 4.8

A force \displaystyle {\mathbf{F}_{1}} = (\displaystyle 9\mathbf{i}+12\mathbf{j}) N and a second force \displaystyle {\mathbf{F}_{2}} = (\displaystyle -4\mathbf{i}+\mathbf{j}) N act on a particle.

(a) Find the magnitude of \displaystyle {\mathbf{F}_{1}}.


(b) Draw a diagram to show the direction of \displaystyle {\mathbf{F}_{1}}, and calculate the angle between \displaystyle {\mathbf{F}_{1}} and \displaystyle \mathbf{i}.


(c) Find the resultant of \displaystyle {\mathbf{F}_{1}} and \displaystyle {\mathbf{F}_{2}}.


Answer a
Answer 4.8a
Answer b
Answer 4.8b
Answer c
Answer 4.8c
Solution a
Solution 4.8a
Solution b
Solution 4.8b
Solution c
Solution 4.8c


Exercise 4.9

Four forces are shown on the diagram below.

Image:E4.9.GIF

a) Find the magnitude of the resultant of the four forces.

b) Find the angle between the resultant force and the unit vector \displaystyle \mathbf{i} correct to the nearest degree.


Answer
Answer 4.9
Solution a
Solution 4.9a
Solution b
Solution 4.9b
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