A simple pendulum is made from a Bob of mass 0.04kg suspended on a light spring of length 1.4m. keeping the string taut pendulum is pulled to one side until it has gained a height 0.1m. Calculated
A. the total energy of oscillation?
B. the amplitude of the resulting oscillation?
C . the period of the resulting oscillations?
D.the maximum velocity of the bob?
E.the maximum kinetic energy?
energy = m g h = 0.04 * 9.81 * 0.1 Joules
L - L cos A = .1 m
1.4 (1-cos A) = .1 m
1 - cos A = .0714
cos A = about 1 - A^2/2 for small angle A
A^2/2 = .0714
A^2 = .1428
A = 0.378 radians amplitude
(1/2) m v^2 at bottom = (1/2)m (omega* L)^2 = energy from part A
etc
To calculate the total energy of the oscillation, we need to consider both potential energy and kinetic energy.
A. Total Energy:
The total energy, E, of the oscillation is given by the sum of potential energy (PE) and kinetic energy (KE), which can be expressed as:
E = PE + KE
1) Potential Energy:
The potential energy, PE, of the pendulum is given by the formula:
PE = m * g * h
Where:
m = mass of the bob (0.04 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height gained by the pendulum (0.1 m)
Substituting the values into the formula:
PE = 0.04 kg * 9.8 m/s^2 * 0.1 m
PE = 0.0392 J
2) Kinetic Energy:
The kinetic energy, KE, of the pendulum is given by the formula:
KE = (1/2) * m * v^2
Where:
m = mass of the bob (0.04 kg)
v = velocity of the bob
Since the pendulum is pulled to one side and then released, at the highest point of the oscillation, the velocity of the bob is zero. Therefore, at this point, the kinetic energy is also zero.
Thus, the total energy of the oscillation is equal to the potential energy:
E = PE
E = 0.0392 J
Therefore, the total energy of the oscillation is 0.0392 J.
B. Amplitude:
The amplitude of the resulting oscillation can be determined by the height gained by the pendulum.
The amplitude, A, is half of the total displacement, which is equal to the height gained:
A = h/2
A = 0.1 m / 2
A = 0.05 m
Therefore, the amplitude of the resulting oscillation is 0.05 m.
C. Period:
The period, T, of the resulting oscillations can be calculated using the formula:
T = 2 * π * √(l/g)
Where:
l = length of the spring (1.4 m)
g = acceleration due to gravity (9.8 m/s^2)
Substituting the values into the formula:
T = 2 * π * √(1.4 m / 9.8 m/s^2)
T ≈ 2 * 3.14 * √(0.1429 s^2)
T ≈ 2.84 s
Therefore, the period of the resulting oscillations is approximately 2.84 seconds.
D. Maximum Velocity:
The maximum velocity occurs at the lowest point of the oscillation. The velocity at the lowest point is given by the formula:
v = √(2 * g * h)
Where:
g = acceleration due to gravity (9.8 m/s^2)
h = height gained by the pendulum (0.1 m)
Substituting the values into the formula:
v = √(2 * 9.8 m/s^2 * 0.1 m)
v ≈ √(1.96 m^2/s^2)
v ≈ 1.40 m/s
Therefore, the maximum velocity of the bob is approximately 1.40 m/s.
E. Maximum Kinetic Energy:
The maximum kinetic energy occurs at the lowest point of the oscillation. This is when the potential energy is equal to zero.
Therefore, the maximum kinetic energy is equal to the total energy of the oscillation:
KE = E
KE = 0.0392 J
Therefore, the maximum kinetic energy of the bob is 0.0392 J.
To answer these questions, we can use the principles of conservation of energy in simple harmonic motion.
A. The total energy of oscillation is the sum of potential energy and kinetic energy. In this case, the initial potential energy is equal to the gravitational potential energy gained by lifting the pendulum. The potential energy is given by:
Potential Energy = mass * gravity * height
where mass is 0.04kg, gravity is approximately 9.8 m/s², and height is 0.1m.
B. The amplitude of the resulting oscillation is equal to the maximum displacement from the equilibrium position. In this case, the amplitude is equal to the initial height, which is 0.1m.
C. The period of an ideal simple pendulum is given by the formula:
Period = 2π * √(length / gravity)
where length is 1.4m and gravity is approximately 9.8 m/s².
D. The maximum velocity of the bob occurs at the equilibrium position and can be calculated using the formula:
Maximum velocity = amplitude * angular frequency
The angular frequency (ω) is given by:
Angular frequency = 2π / period
E. The maximum kinetic energy occurs at the equilibrium position and is given by:
Maximum kinetic energy = 0.5 * mass * maximum velocity^2
Now let's calculate the answers to these questions:
A. Total Energy = Potential Energy + Kinetic Energy
Potential Energy = 0.04kg * 9.8 m/s² * 0.1m
Kinetic Energy = 0 (since the pendulum is initially at rest)
Total Energy = Potential Energy
= 0.04kg * 9.8 m/s² * 0.1m
B. Amplitude = 0.1m
C. Period = 2π * √(1.4m / 9.8 m/s²)
D. Maximum velocity = Amplitude * Angular frequency
= 0.1m * (2π / Period)
E. Maximum kinetic energy = 0.5 * 0.04kg * (Maximum velocity)^2
By substituting the given values into the equations, we can solve for the answers to each question.