a simple pendulum of lengh l,the mass of whose bob is m,is observed to have a speed v0 when the cord makes an angle theta0(0<theta<pi/2).In terms of g and the foregoing given quantities,determinate the least value v2 that v0 could have if the cord is to achieve a horizontal position during the motion, and the speed v3 such that if v0>v3 the pendulum will not oscillate but rather will continue to move around in a verical circle
To find the minimum value of v0 for the cord to achieve a horizontal position during the motion, we need to consider the conservation of mechanical energy.
The mechanical energy of the pendulum consists of its potential energy and kinetic energy. At the highest point of the swing (when the cord is horizontal), the bob has maximum potential energy and no kinetic energy. This potential energy is converted into kinetic energy as the pendulum swings down.
Let's denote the potential energy at the highest point as PE and the initial kinetic energy at angle theta0 as KE. The mechanical energy is given by the sum of PE and KE.
PE = m * g * l * (1 - cos(theta0))
KE = 0.5 * m * v0^2
To achieve a horizontal position, the maximum potential energy at the highest point should be equal to the initial kinetic energy:
PE = KE
m * g * l * (1 - cos(theta0)) = 0.5 * m * v0^2
Simplifying and solving for v0^2:
v0^2 = 2 * g * l * (1 - cos(theta0))
To determine the speed v3 at which the pendulum will stop oscillating and instead move in a vertical circle, we need to consider the conditions for circular motion.
In circular motion, the tension in the cord provides the centripetal force. At the minimum required speed for circular motion, the centripetal force should be equal to the weight of the bob.
Let's denote the required minimum speed as v2 and the tension in the cord as T.
T = m * g
Centripetal force = m * v2^2 / l
Setting these two equal and solving for v2:
m * g = m * v2^2 / l
v2^2 = g * l
To determine the speed v3 at which the pendulum will not oscillate but instead move around in a vertical circle, we need to consider the condition for complete circular motion without any restoring force.
At this speed, the tension in the cord will become zero, and the pendulum will only experience the gravitational force.
T = 0
Centripetal force = m * v3^2 / l = m * g
Simplifying and solving for v3:
v3^2 = g * l
To summarize:
- The least value v0 could have for the cord to achieve a horizontal position during the motion is given by v0^2 = 2 * g * l * (1 - cos(theta0)).
- The minimum speed v2 required for the pendulum to move in a vertical circle is given by v2^2 = g * l.
- The speed v3, at which the pendulum will not oscillate and move in a vertical circle, is given by v3^2 = g * l.