A pendulum, consisting of a bob of mass M on a cord of...

A pendulum, consisting of a bob of mass $M$ on a cord of length $L$, is interrupted in its swing by a peg a distance $d$ below its point of suspension. (a) If the bob is to travel in a full circle of radius $(L-d)$ around the peg, what is the minimum possible speed it can have at the lowest point in its motion, just before it starts to go around? Ignore any decrease in the length of the string due to the peg's circumference. (b) From what minimum angle $\theta$ must the pendulum be released so that the bob attains the speed calculated in (a)? CAN'T COPY THE FIGURE

Key Concepts

Conservation of Energy

This concept involves understanding how energy is transferred between potential energy and kinetic energy in mechanical systems. In the context of the pendulum, the gravitational potential energy lost as the bob descends is converted into kinetic energy, and vice versa as it climbs the circular path around the peg. The conservation principle allows one to relate the speed of the bob at different points in its trajectory based on height differences.

Centripetal Force and Full Circular Motion

For an object to complete a full circle, a sufficient centripetal force must be maintained throughout the motion, especially at the top of the circle where gravitational pulling opposes the necessary centripetal acceleration. This idea establishes a minimum speed requirement at critical points in the motion so that the tension in the string is not exhausted, ensuring that the bob remains in contact with its circular path.

Geometric Relations in Modified Pendulum Systems

When the pendulum's path is altered by a peg, the geometry of motion changes, as the effective length of the string becomes the distance from the peg to the bob. Understanding the relationship between the original string length and the new radius of circular motion is crucial for correctly applying energy conservation and centripetal force requirements to determine the necessary speed and release angle.

Transcript

00:02 Assume we have a pendulum consisting of a chord of length l and a mass m attached to it.

00:11 If this pendulum is allowed to swing and is then interrupted at its lowest point by a peg, the pendulum will transcribe a circle in its motion.

00:25 How do we know what the minimum possible speed at the bottom of its motion, or the pendulum would be to be able to put it.

00:35 Pass through the entire circle.

00:38 Well, let's look at what the conservation of energy tells us.

00:44 Firstly, at the bottom of the circle, this is the initial point of reference.

00:52 We know u, which is the potential energy, mgh, is zero.

01:00 So the bottom of the circle is taken to be at a height zero.

01:07 And the kinetic energy, k, is equal to a half m v i square and this is what we are trying to find v i the velocity as the pendulum begins to make its circular trajectory at the top for our final reference position we know u is equal to m g h and h in this case is just l minus.

01:50 So at this point at the top of the circle, what is the kinetic energy at this point? k is equal to a half mv square the final velocity.

02:11 Well we know that this is centripetal motion.

02:18 So the kinetic energy is m g times r.

02:25 This is due to the fact that this interpidate acceleration g is just v squared divided by r and r being the radius of the circle is just l minus d so that's a half m g into l minus d now the conservation of energy tells us that the energy before has to be the mechanical energy after so a half m v i squared is equal to two mg into l minus d plus a half ng into l minus d.

03:19 So the sum of the kinetic energy and potential energy before, or at the bottom of the circle, has to give us the sum of the potential and kinetic energies at the top of the circle.

03:32 So if we do a bit of rearranging, we find vi the square root of 5 times gravitation acceleration g into the radius of the circle l minus d...

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