A simple pendulem is made from a bob of mass 0.040 kg suspended on a light string of length 1.4 m.keeping taut,the pendulemis pulled to one side until it has gained a height of 0.10 m.calculte
a) the total energy of the oscillation
b) the amplitude of the resulting oscillation
c) the period of theresulting oscillation
d) the maximum velocity of the bob
e) the maximum kenetic energy of the bob
a) really simple, m g h = 0.040 * 9.81 * 0.10 = 0.0392 Joules
If you usually use 10 instead of 9.81 for g It would be just 0.04 Joules
b) well, I guess they mean angle Theta
0.10 meter = 1.4 (1-cos Theta)
cos Theta = 1.3/1.4 = 0.9285
Theta = 21.8 degrees amplitude = 0.380 RADIANS
c. I will assume you know period of pendulum = 2 pi sqrt(L/g)
T = 2 * 3.14 sqrt (1.4 / 9.81) = 2.37 seconds
d. max Ke = (1/2)m v^2 = total E = 0.0392 Joules from a. above
solve for v
e. the same as answer to a. At the bottom all the energy is kinetic.
a) The total energy of the oscillation can be calculated by summing the potential energy and the kinetic energy of the bob.
The potential energy of the bob at its maximum height can be calculated using the formula: PE = m*g*h
Where m is the mass of the bob (0.040 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (0.10 m).
PE = 0.040 kg * 9.8 m/s^2 * 0.10 m = 0.0392 J
The kinetic energy at the extreme position (highest point) is zero as the velocity is zero.
Therefore, the total energy of the oscillation would be equal to the potential energy at the maximum height:
Total Energy = 0.0392 J
b) The amplitude of the resulting oscillation can be determined by halving the height the bob was pulled to one side.
Amplitude = 0.10 m / 2 = 0.05 m
c) The period of the resulting oscillation can be calculated using the formula: T = 2π√(L/g)
Where L is the length of the string (1.4 m) and g is the acceleration due to gravity (9.8 m/s^2).
T = 2π√(1.4 m / 9.8 m/s^2) ≈ 3.0 s
d) The maximum velocity of the bob can be determined using the formula: v = ω*A
Where ω is the angular frequency and A is the amplitude.
Since ω = 2π/T, we can substitute the value of T:
v = (2π/T) * A = (2π/3.0 s) * 0.05 m ≈ 0.33 m/s
e) The maximum kinetic energy of the bob can be calculated using the formula: KE = (1/2) * m * v^2
Where m is the mass of the bob (0.040 kg) and v is the maximum velocity (0.33 m/s).
KE = (1/2) * 0.040 kg * (0.33 m/s)^2 ≈ 0.002 J
To calculate the required values for the simple pendulum:
a) To find the total energy of the oscillation, we need to consider both potential energy and kinetic energy.
The potential energy (PE) at the highest point is given by the formula: PE = m * g * h, where m is the mass of the bob, g is the acceleration due to gravity (9.8 m/s^2), and h is the height (0.10 m).
The kinetic energy (KE) at the lowest point is given by the formula: KE = (1/2) * m * v^2, where v is the maximum velocity of the bob when it swings. Since the bob momentarily comes to rest at the highest point, the total energy is equal to the potential energy at the highest point.
Therefore, the total energy of the oscillation (E) can be calculated as follows:
E = PE = m * g * h
= 0.040 kg * 9.8 m/s^2 * 0.10 m
b) The amplitude of the resulting oscillation is equal to the maximum displacement of the bob from its equilibrium position. In this case, as the pendulum was displaced to a height of 0.10 m, the amplitude will also be 0.10 m.
c) The period of the resulting oscillation (T) for a simple pendulum can be calculated using the formula: T = 2π * √(l / g), where l is the length of the string and g is the acceleration due to gravity.
Substituting the given values, we can calculate the period as follows:
T = 2π * √(1.4 m / 9.8 m/s^2)
d) The maximum velocity of the bob occurs at its lowest point, which is equal to the amplitude multiplied by the angular frequency. The angular frequency (ω) can be calculated using the formula: ω = √(g / l).
Therefore, the maximum velocity (v_max) can be calculated as:
v_max = A * ω
v_max = 0.10 m * √(9.8 m/s^2 / 1.4 m)
e) The maximum kinetic energy (KE_max) of the bob occurs when it is at its lowest point. It can be calculated as half the product of the mass of the bob and the square of its maximum velocity.
KE_max = (1/2) * m * v_max^2
= (1/2) * 0.040 kg * (v_max)^2
Using these formulas, we can solve for the requested values.
To calculate the various parameters of the simple pendulum, we can use the equations of motion and the concepts of potential energy and kinetic energy. Let's go step by step:
a) Total Energy of the Oscillation:
The total energy of the oscillation consists of the potential energy (PE) when the bob is at its highest point and the kinetic energy (KE) when the bob is moving fastest. The total energy (TE) is the sum of these two:
TE = PE + KE
Potential energy can be calculated using the formula:
PE = m * g * h
where:
m = mass of the bob (0.040 kg)
g = acceleration due to gravity (9.8 m/s²)
h = height gained by the bob (0.10 m)
Substituting the values into the equation, we have:
PE = 0.040 kg * 9.8 m/s² * 0.10 m = 0.0392 J
To find the kinetic energy, we use the formula:
KE = (1/2) * m * v^2
where v is the velocity of the bob.
Since the bob is at its highest point, the velocity is zero, so the kinetic energy is also zero.
Therefore, the total energy of the oscillation is equal to the potential energy:
TE = PE = 0.0392 J
b) Amplitude of the Resulting Oscillation:
The amplitude of the resulting oscillation is equal to the distance from the equilibrium position to the highest point the pendulum reaches. In this case, it is the same as the height gained by the bob, which is 0.10 m.
c) Period of the Resulting Oscillation:
The period (T) of an ideal simple pendulum can be calculated using the formula:
T = 2π * √(L / g)
where:
L = length of the string (1.4 m)
g = acceleration due to gravity (9.8 m/s²)
Substituting the values, we have:
T = 2π * √(1.4 m / 9.8 m/s²) = 2.82 s
d) Maximum Velocity of the Bob:
The maximum velocity is reached when the bob is passing through its equilibrium position. At this point, all of the potential energy has been converted to kinetic energy. Using the conservation of energy:
PE = KE
m * g * h = (1 / 2) * m * v_max^2
Simplifying the equation, we get:
v_max = √(2 * g * h)
Substituting the values:
v_max = √(2 * 9.8 m/s² * 0.10 m) = 1.40 m/s
e) Maximum Kinetic Energy of the Bob:
The maximum kinetic energy is reached when the bob is at its equilibrium position, so all the potential energy has converted to kinetic energy at this point. Therefore, the maximum kinetic energy is equal to the potential energy at the highest point of the oscillation:
KE_max = PE = 0.0392 J
To summarize:
a) The total energy of the oscillation is 0.0392 J.
b) The amplitude of the resulting oscillation is 0.10 m.
c) The period of the resulting oscillation is 2.82 s.
d) The maximum velocity of the bob is 1.40 m/s.
e) The maximum kinetic energy of the bob is 0.0392 J.