A pendulum consists of a 2.0 kg stone swinging on a 4.0 m string of negligible mass. The stone

A pendulum consists of a 2.0 kg stone swinging on a 4.0 m...

A pendulum consists of a $2.0 \mathrm{~kg}$ stone swinging on a 4.0 $\mathrm{m}$ string of negligible mass. The stone has a speed of $8.0 \mathrm{~m} / \mathrm{s}$ when it passes its lowest point. (a) What is the speed when the string is at $60^{\circ}$ to the vertical? (b) What is the greatest angle with the vertical that the string will reach during the stone's motion? (c) If the potential energy of the pendulum-Earth system is taken to be zero at the stone's lowest point, what is the total mechanical energy of the system?

A pendulum consists of a $2.0 \mathrm{~kg}$ stone swinging on a 4.0 $\mathrm{m}$ string of negligible mass. The stone has a speed of $8.0 \mathrm{~m} / \mathrm{s}$ when it passes its lowest point. (a) What is the speed when the string is at $60^{\circ}$ to the vertical? (b) What is the greatest angle with the vertical that the string will reach during the stone's motion?
(c) If the potential energy of the pendulum-Earth system is taken to be zero at the stone's lowest point, what is the total mechanical energy of the system?

Key Concepts

Kinetic Energy

Kinetic energy represents the energy a body possesses due to its motion. For a swinging pendulum, this energy is maximized at the lowest point of its swing where the speed is highest and minimized at the turning points where the speed is zero. The conversion between kinetic energy and potential energy governs the pendulum's speed at various positions along its path.

Pendulum Geometry and Dynamics

The motion of a pendulum is governed by its geometrical characteristics, particularly the relationship between the string length and the angle of displacement from the vertical. The vertical displacement can be determined using trigonometric relationships, such as h = L(1 - cos?), which is crucial for translating angular deflections into changes in gravitational potential energy. This geometric relationship underpins the analysis of the pendulum's energy transformations.

Conservation of Mechanical Energy

This principle states that in an isolated system with only conservative forces acting, the total mechanical energy (the sum of kinetic and potential energies) remains constant. In pendulum motion, the energy continuously transforms between kinetic energy (the energy of motion) and gravitational potential energy (the energy due to height), but their total remains unchanged throughout its motion.

Gravitational Potential Energy

Gravitational potential energy in the context of a pendulum is determined by the vertical height of the bob above its lowest position. The energy associated with the gravitational field is calculated by the product of mass, gravitational acceleration, and height (mgh). This concept is essential for defining energy differences as the pendulum swings upward from the lowermost point.

Transcript

00:01 Here in this given problem a stone is tied with a string, a massless string and it is oscillating like a simple pendulum.

00:12 First of all we shift it up to an angle of 60 degree and we have to find its speed there.

00:21 Then we have to find maximum angle at which it will come to rest momentarily before coming back to the initial position, this angle theta that is missing.

00:34 Length of the string that is given to us l, mass of the stone that is also given to us and this is the height h, height of the stone above the reference point, lowest reference point.

00:52 So if the length of the string is l, then this component here it will be given as l cos theta, so this height remaining that will be given by l minus l cos theta or we can say this is h is equal to l bracket 1 minus cos theta.

01:18 Mass of the stone in this pendulum that is given as 2 .0 kilogram, length of the string this is 4 .0 meter, its speed at the lowermost point here, suppose it to be a, it to be b and then here this is c, so va is given as 8 .0 meter per second.

01:49 In the first part of the problem when the angle theta is 60 degree, we know an expression for h, so using in order to find the speed of the stone at point b, using energy conservation of mechanical energy, kinetic energy at point b, half m vb square and gravitational potential energy there mgh that should be equal to initial kinetic energy half m va square.

02:32 Canceling this m from both the sides and this is half vb square is equal to half va square minus gh or in place of h that is l bracket 1 minus cos theta that is 1...

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