The amplitude of a system moving in simple harmonic motion is doubled. Determine the change in
a. The total energy,
b. The maximum speed,
c. The maximum acceleration, and
d. The period.
To determine the changes in total energy, maximum speed, maximum acceleration, and period when the amplitude of a system moving in simple harmonic motion is doubled, we can use the formulas and equations associated with simple harmonic motion. Let's go through each question step by step:
a. Change in total energy:
The total energy of a system in simple harmonic motion is given by the formula:
E = (1/2) mω²A²
Where:
E is the total energy,
m is the mass of the system,
ω is the angular frequency, and
A is the amplitude of the system.
To find the change in total energy, we can compare the original total energy (E1) and the new total energy (E2) when the amplitude is doubled.
If the original amplitude is A, then the new amplitude will be 2A.
Using the formula:
E1 = (1/2) mω₁²A²
E2 = (1/2) mω₂²(2A)²
The angular frequency (ω₁) in the original case is related to the amplitude (A) by the equation:
ω₁ = √(k/m)
Where:
k is the spring constant.
Similarly, the new angular frequency (ω₂) can be determined with the new amplitude (2A):
ω₂ = √(k/m)
Now, substituting the values into the equations, we get:
E1 = (1/2) m(√(k/m))²A² = (1/2) kA²
E2 = (1/2) m(√(k/m))²(2A)² = 2kA²
To find the change (ΔE) in total energy, subtract E1 from E2:
ΔE = E2 - E1 = 2kA² - (1/2) kA² = (3/2) kA²
Therefore, the change in total energy when the amplitude is doubled is (3/2) kA².
b. Change in maximum speed:
The maximum speed of a system in simple harmonic motion is given by the formula:
Vmax = ωA
To find the change in maximum speed, we can compare the original maximum speed (Vmax,1) and the new maximum speed (Vmax,2) when the amplitude is doubled.
Using the angular frequency equations above:
Vmax,1 = √(k/m) * A
Vmax,2 = √(k/m) * (2A)
To find the change (ΔVmax) in maximum speed, subtract Vmax,1 from Vmax,2:
ΔVmax = Vmax,2 - Vmax,1 = √(k/m) * (2A) - √(k/m) * A = √(k/m) * A
Therefore, the change in maximum speed when the amplitude is doubled is √(k/m) * A.
c. Change in maximum acceleration:
The maximum acceleration of a system in simple harmonic motion is given by the formula:
amax = ω²A
To find the change in maximum acceleration, we can compare the original maximum acceleration (amax,1) and the new maximum acceleration (amax,2) when the amplitude is doubled.
Using the angular frequency equations above:
amax,1 = (√(k/m))² * A
amax,2 = (√(k/m))² * (2A)
To find the change (Δamax) in maximum acceleration, subtract amax,1 from amax,2:
Δamax = amax,2 - amax,1 = (√(k/m))² * (2A) - (√(k/m))² * A = (√(k/m))² * A
Therefore, the change in maximum acceleration when the amplitude is doubled is (√(k/m))² * A.
d. Change in period:
The period of a system in simple harmonic motion is the time it takes to complete one full cycle and is given by the formula:
T = 2π√(m/k)
To find the change in period, we can compare the original period (T1) and the new period (T2) when the amplitude is doubled.
For the given formulas, the period is not directly dependent on the amplitude, so the period remains the same. Therefore, there is no change in the period when the amplitude is doubled.
In summary:
a. The change in total energy is (3/2) kA².
b. The change in maximum speed is √(k/m) * A.
c. The change in maximum acceleration is (√(k/m))² * A.
d. There is no change in the period when the amplitude is doubled.