A pendulum bob of mass -m is suspended at rest a constant horizontal force f= mg starts acting on it . Find thecmaximum deflection of the string and the maximum tension in the string
To find the maximum deflection of the string and the maximum tension in the string, we need to consider the forces acting on the bob and apply the principles of circular motion.
Let's analyze the forces acting on the pendulum bob:
1. Gravitational force (mg): This force acts vertically downward and is equal to the product of the mass (m) and the acceleration due to gravity (g).
2. Tension in the string (T): This force acts along the string and provides the necessary centripetal force for circular motion.
Now, let's consider the motion of the pendulum bob:
When the horizontal force (f = mg) starts acting on the bob, it experiences a horizontal acceleration. This acceleration causes the bob to move in a circular path instead of simply hanging vertically.
The maximum deflection of the string occurs when the tension in the string is maximum and provides the centripetal force required to keep the bob in circular motion.
To find the maximum deflection of the string, we need to calculate the horizontal acceleration of the bob using the horizontal force:
Force = mass × acceleration
mg = (-m) × a (since the force is acting horizontally in the negative direction)
By canceling out the mass on both sides, we get:
g = -a
Therefore, the horizontal acceleration of the bob is -g.
The maximum deflection of the string can be calculated using the formula for centripetal acceleration:
a_c = r × ω^2
Since we are interested in the maximum deflection, we can assume that the pendulum bob will move in a semi-circular path (arm of length 'r').
The angular velocity (ω) can be found by relating it to the horizontal acceleration:
a_c = r × ω^2
-g = r × ω^2
Simplifying the equation, we have:
ω^2 = -g / r
Now, the maximum deflection is given by:
Maximum Deflection = 2r
To find the maximum tension in the string, we need to remember that the tension (T) provides the necessary centripetal force. Using the formula for centripetal force:
F_c = m × a_c
Since we know the horizontal acceleration (a_c = -g) and the mass (m), we can find the maximum tension:
F_c = m × (-g)
T = m × (-g)
Therefore, the maximum tension in the string is equal to the mass multiplied by the acceleration due to gravity.
To summarize:
- The maximum deflection of the string is twice the length of the arm (2r).
- The maximum tension in the string is equal to the product of the mass (m) and the acceleration due to gravity (-g).