**Forces in equilibrium**

1. An object is said to be in a state of equilibrium when the object is :

a) At rest (static equilibrium)

b) Moving with uniform velocity (dynamic equilibrium)

Examples:

a) At rest (static equilibrium)

- A book at rest on the table is in a state of equilibrium because there is no unbalanced force acting on the book.

- There are two forces acting on the book :

a) The weight of the book

b) The normal reaction on the table.

- The weight of the book is balanced by the normal reaction of the table. There is no unbalanced force acting on the book.

- An object is said to be in the state of equilibrium.

b) Moving with uniform velocity (dynamic equilibrium)

- An air puck moving with uniform velocity.

- The forces acting on the air puck are the weight and the normal reaction of the table.

- As the track is smooth, there is no frictional force acting against the motion of the air puck.

- There is no unbalanced forced acting on the air puck. The weight of the air puck is balanced by the normal reaction of the table.

- An object is said to said to be in a state of equilibrium.

Problem : The picture hanging on the wall is said to be in the state of equilibrium.

a) Another way of determining the net force acting on the picture frame involves the resolution of forces into components that are perpendicular to each other.

b) T1 is resolved into its components T1 sin Ɵ and T1 cos Ɵ that are perpendicular to each other. Similarly, T2 is resolved into its components T2 sin Ɵ and T2 cos Ɵ.

c)Since the picture frame does not move, it is in a state of equilibrium.

d) So, the resultant of the horizontal component is zero;

T1 cos Ɵ - T2 cos Ɵ = 0

The resultant of the vertical component must be zero also;

T1 sin Ɵ + T2 sin Ɵ - W = 0

Problem : The picture frame is hung this way with the tension of strings T1 and T2 making an angle Ɵ with the frame.

Problem : Ahmad and Zaki are pulling a boat along the river. Each of them exerts a force of 600 N directed at a 30

a) The forces F1 and F2 are drawn as below.

**To Find Resultant Force**

**Example 1**

Method 1 : Tail to head triangle of forces

a) The tension of the string that hangs the picture are T1 and T2. The string is at angle Ɵ from thee picture frame and the weight of the frame is W.

b) With the length of an arrow representing the magnitude of a force and the direction of an arrow representing the direction of a force, the resultant of the forces acting on an object can be obtained.

c) When the three scaled arrows representing the forces T1 and T2 and W are drawn from end to end, they form a closed triangle.

d) Therefor, the forces T1, T2 and W are said to be in equilibrium. The vector addition of these three forces will be a net force of 0 N.

Method 2 : Resolution of forces

a) Another way of determining the net force acting on the picture frame involves the resolution of forces into components that are perpendicular to each other.

b) T1 is resolved into its components T1 sin Ɵ and T1 cos Ɵ that are perpendicular to each other. Similarly, T2 is resolved into its components T2 sin Ɵ and T2 cos Ɵ.

c)Since the picture frame does not move, it is in a state of equilibrium.

d) So, the resultant of the horizontal component is zero;

T1 cos Ɵ - T2 cos Ɵ = 0

**T1 cos****Ɵ = T2 cos Ɵ**The resultant of the vertical component must be zero also;

T1 sin Ɵ + T2 sin Ɵ - W = 0

**T1 sin****Ɵ + T2 sin Ɵ = W****Example2**

Problem : The picture frame is hung this way with the tension of strings T1 and T2 making an angle Ɵ with the frame.

Method : Resolution of Forces

Since it does not moves, it is in a state of equilibrium. The net forces acting on it should be zero.

Horizontal force,

T1 cos Ɵ - T2 cos Ɵ = 0

**T1 cos****Ɵ = T2 cos Ɵ**Vertical force,

T1 sin Ɵ + T2 sin Ɵ - W = 0

**T1 sin****Ɵ + T2 sin Ɵ = W****Example 3**Problem : Ahmad and Zaki are pulling a boat along the river. Each of them exerts a force of 600 N directed at a 30

^{o}angle relative to the forward motion of the boat. The boat moves with constant velocity. What is the Resultant Force, F, acting on the boat?Method : Paralellogram

a) The forces F1 and F2 are drawn as below.

b) By completing the parallelogram, the resultant force, F, is represented by the diagonal of the parallelogram. As the result, the boat moves forward in the direction of the resultant force.

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