Problem:
A ball weighing 100N is suspended vertically by a string 5 m long. Find the magnitude of the force which should be applied horizontally to hold the ball one meter above the lowest point. Also determine the tensions in the string.

Solution:
To find the magnitude of the force required to hold the ball one meter above the lowest point, we need to consider the forces acting on the ball and the equilibrium conditions.

Forces Acting on the Ball:
1. Weight (W): The weight of the ball is acting vertically downwards and its magnitude is given as 100N.
2. Tension in the string (T): The tension in the string acts along the string and its magnitude varies with the position of the ball.

Equilibrium Conditions:
For the ball to be in equilibrium, the net force acting on it should be zero. This implies that the vertical component of the tension in the string should balance the weight of the ball, and the horizontal component of the tension should be equal to the force required to hold the ball one meter above the lowest point.

Calculating the Tension in the String:
Let's assume the angle between the string and the vertical direction is θ. The tension in the string can be calculated using the following equations:

1. Vertical Component of Tension (Tvertical):
The vertical component of the tension balances the weight of the ball. Therefore, Tvertical = Weight = 100N.

2. Horizontal Component of Tension (Thorizontal):
The horizontal component of the tension is equal to the force required to hold the ball one meter above the lowest point. Let's calculate it.

Using Trigonometry:
We can use trigonometric ratios to find the horizontal component of tension. In a right-angled triangle formed by the string, the horizontal component is given by:

Thorizontal = T * sin(θ)

In this case, the length of the string is 5m, and we want to hold the ball 1m above the lowest point. Therefore, the vertical height (h) is 4m.

Using the Pythagorean theorem:
h^2 + 1^2 = 5^2
16 + 1 = 25
h = √(25-1)
h = √24
h = 2√6

Now, we can calculate the angle θ using trigonometric ratios:
sin(θ) = h / string length
sin(θ) = 2√6 / 5
θ = sin^(-1)(2√6 / 5)

Substituting the value of θ in the equation for Thorizontal:
Thorizontal = T * sin(θ)

Summary:
- The magnitude of the force required to hold the ball one meter above the lowest point is equal to the horizontal component of the tension in the string.
- The tension in the string can be calculated using trigonometric ratios, considering the vertical and horizontal components of the tension.
- The vertical component of the tension balances the weight of the ball, and the horizontal component is equal to the force required to hold the ball one meter above the lowest point.
- To calculate the horizontal component of the tension, we need to find the angle between the string and the vertical direction using trigonometry.
- Substituting the