an object of mass 10kg is whirled round in a horizontal circle of radius 4metres by a revolving string which is inclined to vertical. if the uniform speed of the object is 5metres per second. 1. calculate the tension in the string. 2. the angle of
of the string to the vertical.
To calculate the tension in the string, we can use the centripetal force formula:
Fc = (m * v^2) / r
where,
Fc is the centripetal force,
m is the mass of the object,
v is the speed of the object, and
r is the radius of the circular path.
Given:
m = 10 kg
v = 5 m/s
r = 4 m
1. Calculate the tension in the string:
Fc = (m * v^2) / r
Fc = (10 kg * (5 m/s)^2) / 4 m
Fc = (10 * 25) / 4
Fc = 250 / 4
Fc = 62.5 N
Therefore, the tension in the string is 62.5 Newtons (N).
2. To find the angle of the string to the vertical, we can use trigonometry. Let's call this angle θ.
sinθ = opposite/hypotenuse
sinθ = r / L
where,
L is the length of the string, and
r is the radius of the circular path.
We can find L using the Pythagorean theorem:
L^2 = r^2 + h^2
where,
h is the vertical height from the center of the circle to the string.
Since the string is inclined to the vertical, we can assume that the vertical height h is equal to r*cosθ.
L^2 = r^2 + (r*cosθ)^2
L^2 = r^2 + r^2*cos^2(θ)
L^2 = r^2 * (1 + cos^2(θ))
L = r * sqrt(1 + cos^2(θ))
Now, let's substitute the values:
L = 4 * sqrt(1 + cos^2(θ))
Since sinθ = r/L, we have:
sinθ = r / (4 * sqrt(1 + cos^2(θ)))
sin^2(θ) = r^2 / (16 * (1 + cos^2(θ)))
sin^2(θ) = 1 / (16 / r^2 * (1 + cos^2(θ)))
sin^2(θ) = 1 / (16 / 16 * (1 + cos^2(θ)))
sin^2(θ) = 1 / (1 + cos^2(θ))
1 - cos^2(θ) = 1 + cos^2(θ)
2cos^2(θ) = 0
cos^2(θ) = 0
cos(θ) = 0
θ = 90 degrees
Therefore, the angle of the string to the vertical is 90 degrees.
To find the tension in the string and the angle of the string to the vertical, we can use Newton's laws of motion and the concept of centripetal force.
1. Calculating the tension in the string:
In this case, the tension in the string acts as the centripetal force keeping the object moving in a circular path. The formula for centripetal force is given by:
F = (m * v^2) / r
where F is the centripetal force, m is the mass of the object, v is the velocity, and r is the radius of the circular path.
In this scenario, the mass of the object is 10 kg, and the velocity is 5 m/s. The radius is given as 4 meters. Substituting these values into the formula:
F = (10 kg * (5 m/s)^2) / 4 m = 125 N
Therefore, the tension in the string is 125 Newtons.
2. Calculating the angle of the string to the vertical:
Since the string is revolving in a horizontal circle, the tension in the string has both a vertical component (mg) and a horizontal component (F). The angle between the string and the vertical can be found using trigonometry.
tanθ = (horizontal component of tension) / (vertical component of tension)
In this case, the horizontal component of tension is equal to the centripetal force (125 N), and the vertical component is equal to the weight of the object (mass * gravitational acceleration).
The weight (mg) = 10 kg * 9.8 m/s^2 = 98 N
Substituting these values into the formula:
tanθ = 125 N / 98 N
Taking the arctan of both sides to find the angle:
θ = arctan(125 N / 98 N)
Using a calculator, we find θ ≈ 51.04 degrees.
Therefore, the angle of the string to the vertical is approximately 51.04 degrees.
horizontal length: 4sinTheta
horizontal force= 10*v^2/r= 10*(5/4sinTheta)^2
vertical force= 10*9.8
TanTheta= 10*9.8/(5/4sinTheta)^2
if the angles are small, then tanTheta=sinTheta
TanTheta=98*16Tan^2Theta/25
or 1/tanTheta=98*16/25
tanTheta= = 0.0159438776
Theta= .91 deg