an object of mass m is suspended by a string of length 2m .the object is rotated in a horizontal circle show the relationship between the tension and mass and linear velocity of the body when the string make an angle to the horizontal

When an object of mass m is suspended by a string of length 2m and rotated in a horizontal circle, the tension in the string and the linear velocity of the body are related to the angle the string makes with the horizontal.

Let's denote the angle the string makes with the horizontal as θ.

1. Tension in string:
The tension in the string can be found using the centripetal force requirement for circular motion. As the object is rotating in a horizontal circle, the centripetal force is provided by the tension in the string.

The centripetal force (Fc) is given by:
Fc = m * (v^2 / r)

Where:
m is the mass of the object
v is the linear velocity of the body
r is the radius of the circular path (which is equal to the length of the string, 2m)

Since the tension in the string provides the centripetal force, the tension (T) is equal to Fc:

T = Fc

2. Linear velocity of the body:
The linear velocity of the body can be found using the formula:

v = ω * r

Where:
v is the linear velocity of the body
ω is the angular velocity of the object
r is the radius of the circular path (length of the string, 2m)

The angular velocity (ω) is related to the string angle (θ) through the formula:

ω = √(g / r) * tan(θ)

Where:
g is the acceleration due to gravity (approximately 9.8 m/s^2)

Substituting the value of ω in the formula for v, we get:

v = √(g / r) * tan(θ) * r

Simplifying,

v = √(g * r) * tan(θ)

So, the tension in the string (T) is related to the mass (m) and the linear velocity (v) of the body when the string makes an angle to the horizontal.

To understand the relationship between the tension in the string, the mass of the object, and the linear velocity of the object when the string makes an angle with the horizontal, we consider the forces acting on the object.

1. Gravity: The weight of the object acts vertically downward and can be given as W = mg, where m is the mass of the object and g is the acceleration due to gravity.

2. Tension: The tension in the string is the force applied by the string on the object to keep it in circular motion. Let's denote the tension as T.

3. Centripetal force: The tension in the string provides the centripetal force required to keep the object moving in a circle. The centripetal force can be given as Fc = (mv²)/r, where m is the mass of the object, v is the linear velocity of the object, and r is the radius of the circular path (equal to the length of the string).

When the string makes an angle with the horizontal, let's call it θ, the tension (T) can be resolved into two components: one along the vertical direction (Tcosθ) and one along the horizontal direction (Tsinθ).

The vertical component (Tcosθ) balances the weight of the object (mg), while the horizontal component (Tsinθ) provides the centripetal force (Fc).

Equating these forces, we have:

Tcosθ = mg (1)

Tsinθ = (mv²)/r (2)

Solving equation (1) for T, we get:

T = mg/cosθ

Plugging this into equation (2), we have:

mg*sinθ/cosθ = (mv²)/r

Simplifying further, we get:

tanθ = (v²)/rg

This equation shows the relationship between the angle (θ), linear velocity (v), and the radius of the circular motion (r). The tension in the string (T) can be determined using T = mg/cosθ.

We can observe that the tension in the string is independent of the linear velocity of the object. It depends only on the mass of the object and the angle between the string and the horizontal.