A simple pendulum is made from a bob of mass 0.04 kg suspended on alight string of length 1.4m. Keeping the string taut, the pendulum is pulled to one side until it has gained a height of 0.1m. calculate:
a. The total energy of the oscillation
(b) the amplitude of the resulting oscillation
(c) the period of the resulting oscillation
(d) the maximum velocity of the bob
(e) the maximum kinetic energy of the bob
(a) M g H, where H = 0.1 m, M = mass and g = acceleration of gravity
(b) If amplitude (A) is the maximum horizontal displacement, then
A = sqrt[1.4^2 - 1.3^2] = 0.52 m
(c) (2 pi)*sqrt(L/g)
where L = 1.4 m
(d) A*sqrt(g/L)
(e) M g H (same as maximum energy)
The time taken by a simple pendulum bob to perform 1000 vibrations is 8 minutes 9 seconds in mumbai and 8 minutes 20 seconds in pune.Calulate the ratio of acceleratio due to gravity in mumbai and pune.
yes
To answer these questions, we need to apply the concepts of energy and simple harmonic motion in a pendulum. We can break down the problem into several parts:
(a) The total energy of the oscillation:
The total energy of the pendulum is the sum of its potential energy and kinetic energy. At the highest point (when the pendulum is pulled to one side), the potential energy is at its maximum, and the kinetic energy is zero. At the lowest point (when the pendulum is at its maximum amplitude), the potential energy is zero, and the kinetic energy is at its maximum.
To calculate the potential energy of the pendulum, we can use the formula:
Potential Energy = m * g * h
where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
Potential Energy = 0.04 kg * 9.8 m/s^2 * 0.1 m
Potential Energy = 0.04 J
Since the potential energy is at its maximum at the highest point, the total energy is also 0.04 J.
(b) The amplitude of the resulting oscillation:
The amplitude of a pendulum is the maximum displacement from its equilibrium position. In this case, the pendulum is pulled to one side until it has gained a height of 0.1 m. Therefore, the amplitude is 0.1 m.
(c) The period of the resulting oscillation:
The period of a pendulum is the time it takes to complete one full oscillation, back and forth. It can be calculated using the formula:
Period = 2 * π * √(L / g)
where L is the length of the pendulum and g is the acceleration due to gravity.
Period = 2 * π * √(1.4 m / 9.8 m/s^2)
Period ≈ 3.01 s
(d) The maximum velocity of the bob:
The maximum velocity of the bob occurs at the lowest point (maximum amplitude). The maximum velocity can be calculated using the equation:
Velocity = √(2 * g * h)
where g is the acceleration due to gravity and h is the maximum displacement (amplitude).
Velocity = √(2 * 9.8 m/s^2 * 0.1 m)
Velocity ≈ 1.98 m/s
(e) The maximum kinetic energy of the bob:
The maximum kinetic energy occurs when the potential energy is zero and the kinetic energy is at its maximum. The kinetic energy can be calculated using the formula:
Kinetic Energy = 0.5 * m * v^2
where m is the mass and v is the velocity.
Kinetic Energy = 0.5 * 0.04 kg * (1.98 m/s)^2
Kinetic Energy ≈ 0.078 J
So, the maximum kinetic energy of the bob is approximately 0.078 J.