8.8 Resolution of forces (application)

Questions and their answers are presented here in the module text format as if it were an extension of the treatment of the topic. The idea is to provide a verbose explanation, detailing the application of theory. Solution presented is, therefore, treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.

Hints for solving problems

We resolve a force along the axes of a coordinate system (see Components of a vector ) in following manner :

1 : Select coordinate system such that maximum numbers of forces are along the axes of chosen coordinate system.

2 : Determine “x” and “y” components of force by considering acute angle between the direction of axis and force.

3 : Use cosine of the acute angle for the component of axis with which angle is measured. Use sine of the angle for the axis perpendicular to the other axis. If “θ” be the angle that the force vector makes with x – axis, then components along “x” and “y” axes are :

$$F_x=F\cos \theta$$

$$F_y=F\sin \theta$$

4 : If the component i.e. projection of force is in the opposite direction of the reference axis, then we prefix the component with a negative sign.

Representative problems and their solutions

We discuss problems, which highlight certain aspects of the study leading to the analysis framework for law of motion. The questions are categorized in terms of the characterizing features of the subject matter :

• Balanced forces
• Unbalanced forces
• Acceleration due to gravity
• Forces on an incline

Balanced forces

Problem 1 : Two small spherical objects of mass “m” and “M” are attached with strings as shown in the figure. Find the angle “θ” such that the given system is in equilibrium.

Solution : Every point of the system is under action of balanced force system. We shall work through the points "A" and "B", where masses are attached.

Let “ $T_1$ ” and “ $T_2$ ” be the tensions in the two strings, meeting at point "A". The forces at "A" are shown in the figure above. Considering balancing of forces in "x" and "y" directions :

$$\sum F_x=T_1\cos {45}^0-T_2\cos {45}^0=0$$

$$\Rightarrow T_1=T_2=T\left(\mathrm{say}\right)$$

$$\sum F_y=T_1\sin {45}^0+T_1\sin {45}^0=mg$$

$$\Rightarrow T\sin {45}^0+T\sin {45}^0=mg$$

$$\Rightarrow \frac{2T}{\sqrt{2}}=mg$$

$$\Rightarrow T=\frac{mg}{\sqrt{2}}$$

Let “ $T_3$ ” be the tension in the upper section. The forces on the mass “M” is shown in the figure above. Considering balancing of forces in "x" and "y" directions :

$$\sum F_x=T_3\cos \theta -T\sin {45}^0=0$$

Substituting for “T” and evaluating, we have :

$$\Rightarrow T_3\cos \theta =\frac{mg}{\sqrt{2}}X\frac{1}{\sqrt{2}}=\frac{mg}{2}$$

$$\sum F_y\Rightarrow T_3\sin \theta =T\cos {45}^0+Mg$$

Substituting for "T",

$$\Rightarrow T_3\sin \theta =\frac{T}{\sqrt{2}}+Mg=\frac{mg}{\sqrt{2}}X\frac{1}{\sqrt{2}}+Mg$$

$$\Rightarrow T_3\sin \theta =\frac{\mathrm{mg}}{2}+Mg$$

Taking ratio of the resulting equations in the analysis of forces on "M",

$$\tan \theta =\frac{\frac{mg}{2}+Mg}{\frac{mg}{2}}$$

$$\Rightarrow \tan \theta =\left(1+\frac{2M}{m}\right)$$

$$\Rightarrow \theta =\tan {}^{-1}\left(1+\frac{2M}{m}\right)$$

Unbalanced forces

Problem 2 : A force “F” produces an acceleration “a” when applied to a body of mass “m”. Three coplanar forces of the same magnitudes are applied on the same body simultaneously as shown in the figure. Find the acceleration of the body.

Solution : We select two axes of coordinate system so that they align with two mutually perpendicular forces as shown in the figure. We keep in mind that three forces are coplanar.

Taking components of forces in “x” and “y” directions,

$$\sum F_x=F-F\cos {30}^0=F\left(1-\frac{\sqrt{3}}{2}\right)=F\left(\frac{2-\sqrt{3}}{2}\right)$$

$$\sum Fy=F-F\sin {30}^0=F\left(1-\frac{1}{2}\right)=\frac{F}{2}$$

The net force is given by :

$$F_{net}=\sqrt{\left(F_x^2+F_y^2\right)}=\frac{F}{2}\sqrt{\{1+{\left(2-\sqrt{3}\right)}^2\}}$$

$$\Rightarrow F_{net}=\frac{F}{2}\sqrt{\left(1+4+3-4\sqrt{3}\right)}$$

$$\Rightarrow F_{\mathrm{net}}=\frac{F}{2}\sqrt{\left(8-4\sqrt{3}\right)}$$

$$\Rightarrow \sum F_{\mathrm{net}}=F\sqrt{\left(2-\sqrt{3}\right)}$$

But, it is given that F = ma. Substituting for "F" in the equation, we have :

$$\Rightarrow F_{\mathrm{net}}=ma\sqrt{\left(2-\sqrt{3}\right)}$$

Let acceleration of the body under three coplanar forces be a’. Then,

$$\Rightarrow ma\prime =ma\sqrt{\left(2-\sqrt{3}\right)}$$

$$\Rightarrow a\prime =a\sqrt{\left(2-\sqrt{3}\right)}$$

Acceleration due to gravity

Problem 3 : A small cylindrical object slides down the smooth groove of 10 m on the surface of an incline plane as shown in the figure. If the object is released from the top end of the groove, then find the time taken to travel down the length.

Solution : In order to find the time taken to travel down the incline, we need to know acceleration along the groove. The object travels under the influence of gravity. The component of acceleration due to gravity along the incline (GE), as shown in the figure below, is :

$$a\prime =g\cos {60}^0$$

This component of acceleration makes an angle 60° with the groove. Hence, component of acceleration along the groove (CA) is given as :

$$a=a\prime \cos {60}^0$$

Combining two equations, the acceleration along the groove, "a", is :

$$a=g\cos {60}^0\cos {60}^0=gX\frac{1}{2}X\frac{1}{2}=\frac{g}{4}$$

The acceleration along the groove is constant. As such, we can apply equation of motion for constant acceleration :

$$x=ut+\frac{1}{2}a{t}^2$$

$$\Rightarrow 10=0+\frac{1}{2}X\frac{g}{4}X{t}^2$$

$$\Rightarrow t=\sqrt{\left(\frac{80}{g}\right)}=\sqrt{8}=2\sqrt{2}s$$

Forces on an incline

Problem 4 : Four forces act on a block of mass “m”, placed on an incline as shown in the figure. Then :

• Resolve forces along parallel and perpendicular to incline and find net component forces in two directions.
• Resolve forces along horizontal and vertical directions and find net component forces in two directions.

Solution : The free body diagram with four forces with coordinates are as shown in the figure below.

The net component of forces along two axes are :

$$\sum F_x=F\cos \alpha -F_F-mg\sin \theta$$

$$\sum Fy=N+F\sin \alpha -mg\cos \theta$$

Now, we select “x” and “y” axes along horizontal and vertical directions as shown. Note the angles that different forces make with the axes.

The net component of forces along two axes are :

$$\sum Fx=F\cos \left(\alpha +\theta \right)-N\sin \theta -F_F\cos \theta$$

$$\sum F_y=N\cos \theta +F\sin \left(\alpha +\theta \right)-mg-F_F\sin \theta$$