A mass 5kg is whirled in a horizontal circle @ 1 end of a string 50cm long,the other end being fixed. If the string when hanging vertically will just support a load of 200kg mass without breaking,find the maximum whirling speed in revolution per second and the maximum angular velocity.
Breaking tensile strength of string = Tmax = 200 kg*9.8 m/s^2 = 1960 N
Then whirling horizontally,
T sin A = M g
T cos A = M V^2/R
T^2 = M^2*[g^2 + (V^2/R)^2]
T = Tmax = M* sqrt [g^2 + (V^2/R)^2]
= 1960 N
Tmax/M = 392 m/s^2
= sqrt [g^2 + (V^2/R)^2]
Solve for V
To find the maximum whirling speed in revolutions per second and the maximum angular velocity, we can use the concept of centripetal force.
1. Calculate the tension in the string when the mass is whirling in a horizontal circle:
Tension = m * g
Tension = 5 kg * 9.8 m/s^2 (acceleration due to gravity)
Tension = 49 N
2. Convert the tension force into centripetal force when the mass is whirling:
Centripetal Force = Tension
Centripetal Force = 49 N
3. Calculate the maximum velocity in m/s using the centripetal force:
Centripetal Force = (mass * velocity^2) / radius
49 N = (5 kg * velocity^2) / 0.5 m (radius in meters)
velocity^2 = (49 N * 0.5 m) / 5 kg
velocity^2 = 4.9 m^2/s^2
velocity = √(4.9) m/s
velocity ≈ 2.21 m/s
4. Convert the maximum velocity into revolutions per second:
Circumference of the circle = 2 * π * radius
Circumference = 2 * π * 0.5 m
Circumference ≈ 3.14 m (approximation)
Revolutions per second = velocity / circumference
Revolutions per second = 2.21 m/s / 3.14 m
Revolutions per second ≈ 0.704 revolutions per second
5. Calculate the maximum angular velocity:
Angular Velocity = Revolutions per second * 2π radians
Angular Velocity = 0.704 revolutions per second * 2π
Angular Velocity ≈ 4.42 radians per second
Therefore, the maximum whirling speed is approximately 0.704 revolutions per second, and the maximum angular velocity is approximately 4.42 radians per second.
To find the maximum whirling speed in revolutions per second and the maximum angular velocity, we need to consider the tension in the string when the mass is whirling and when it is hanging vertically.
Let's start by finding the tension in the string when the mass is hanging vertically.
The tension in the string when the 200kg mass is hanging vertically can be calculated using the formula:
Tension = mass * acceleration due to gravity
Tension = 200kg * 9.8m/s^2
Tension = 1960 N
Now, let's find the tension in the string when the mass is whirling in a horizontal circle.
In circular motion, there is a centripetal force acting towards the center of the circle. This force is provided by the tension in the string. The centripetal force can be calculated using the formula:
Centripetal force = (mass * whirling speed^2) / radius
where radius is the length of the string.
Centripetal force = (5kg * whirling speed^2) / 0.5m
We need to ensure that the tension in the string when whirling does not exceed the tension when hanging vertically. Therefore, the maximum whirling speed can be calculated by equating the centripetal force to the tension when hanging vertically:
(5kg * whirling speed^2) / 0.5m = 1960 N
Simplifying the equation:
whirling speed^2 = (1960 N * 0.5m) / 5kg
whirling speed^2 = 196 m^2/s^2
whirling speed = sqrt(196 m^2/s^2) = 14 m/s
The maximum whirling speed is 14 m/s.
To find the maximum angular velocity, we can use the formula:
Angular velocity = whirling speed / radius
Angular velocity = 14 m/s / 0.5m
Angular velocity = 28 rad/s
Therefore, the maximum angular velocity is 28 rad/s.