Answer:
It will take approximately 2237 swings before the pendulum with length 1.990 m has swung one more swing than the pendulum with length 2.000 m.
Step-by-step explanation:
To calculate the number of swings needed for the pendulum to swing one more swing with the length of 1.990 m compared to 2.000 m, we need to consider the conservation of energy in a simple pendulum system.
Let's first calculate the initial energy of the pendulum when its length is 2.000 m. The potential energy (PE) at the highest point (maximum displacement) is converted into kinetic energy (KE) at the lowest point (bottom of the swing). The total mechanical energy (E) remains constant.
The formula for the period (T) of a simple pendulum is given by:
T = 2π * √(L / g)
where:
L = length of the pendulum
g = acceleration due to gravity (approximately 9.81 m/s²)
Let's calculate the period and initial energy of the pendulum when its length is 2.000 m:
L1 = 2.000 m
g = 9.81 m/s²
T1 = 2π * √(L1 / g)
T1 = 2π * √(2.000 / 9.81)
T1 ≈ 2π * 0.6367
T1 ≈ 3.999 seconds (approx.)
The initial energy (E1) is the sum of potential and kinetic energy:
E1 = PE1 + KE1
At the highest point (maximum displacement), all the energy is potential energy (PE1). At the lowest point (bottom of the swing), all the energy is kinetic energy (KE1).
PE1 = m * g * L1 * (1 - cos(θ_max))
where:
θ_max = maximum angular displacement from the vertical (at the highest point).
Since the pendulum hangs freely, θ_max = 0, and the potential energy at the highest point is zero.
KE1 = 0.5 * m * (V_max)^2
where:
V_max = maximum velocity at the lowest point.
V_max = L1 * (2π / T1)
V_max = 2.000 * (2π / 3.999)
V_max ≈ 3.142 m/s (approx.)
KE1 = 0.5 * 0.200 * (3.142)^2
KE1 ≈ 0.3142 J (approx.)
E1 = 0.3142 J
Now, let's adjust the length of the pendulum to 1.990 m and calculate the energy at the lowest point for one full swing:
L2 = 1.990 m
T2 = 2π * √(L2 / g)
T2 = 2π * √(1.990 / 9.81)
T2 ≈ 2π * 0.6351
T2 ≈ 3.982 seconds (approx.)
V_max_2 = L2 * (2π / T2)
V_max_2 = 1.990 * (2π / 3.982)
V_max_2 ≈ 3.146 m/s (approx.)
KE2 = 0.5 * 0.200 * (3.146)^2
KE2 ≈ 0.3135 J (approx.)
Now, we want to find out how many swings it takes for the pendulum with length 1.990 m to accumulate one more swing than the pendulum with length 2.000 m. Let's call the number of swings needed as "N."
The difference in energy between the two cases is given by:
ΔE = KE2 - KE1
ΔE = 0.3135 J - 0.3142 J
ΔE ≈ -0.0007 J (approx.)
Since the total mechanical energy remains constant, the energy lost in one swing (ΔE) is converted to the potential energy at the highest point for each swing.
The potential energy at the highest point (PE2) is given by:
PE2 = m * g * L2 * (1 - cos(θ_max_2))
where:
θ_max_2 = maximum angular displacement from the vertical (at the highest point) for a swing with length 1.990 m.
Let's calculate θ_max_2:
PE2 = ΔE
m * g * L2 * (1 - cos(θ_max_2)) = -0.0007 J
0.200 * 9.81 * 1.990 * (1 - cos(θ_max_2)) = -0.0007
(1 - cos(θ_max_2)) ≈ -0.0007 / (0.200 * 9.81 * 1.990)
(1 - cos(θ_max_2)) ≈ -0.000018 (approx.)
cos(θ_max_2) ≈ 1 + 0.000018 (approx.)
θ_max_2 ≈ acos(1 + 0.000018) (approx.)
θ_max_2 ≈ 0.0041 radians (approx.)
Now, we know that the pendulum swings from θ_max_2 to -θ_max_2 during each swing. So, the total angular displacement during one swing is 2 * θ_max_2 ≈ 2 * 0.0041 ≈ 0.0082 radians (approx.).
The number of swings needed to accumulate the energy difference is given by:
N = |(ΔE * swings in one swing)| / (PE2 in one swing)
N = |(-0.0007 J * 1 swing)| / (m * g * L2 * (1 - cos(θ_max_2)) * 2)
N ≈ 0.0007 / (0.200 * 9.81 * 1.990 * (1 - cos(0.0082))) (approx.)
Now, calculate N:
N ≈ 0.0007 / (0.200 * 9.81 * 1.990 * (1 - cos(0.0082)))
N ≈ 0.0007 / (0.200 * 9.81 * 1.990 * (1 - 0.99992))
N ≈ 0.0007 / (0.200 * 9.81 * 1.990 * 0.00008)
N ≈ 0.0007 / 0.00031296
N ≈ 2237.18 (approx.)
So, it will take approximately 2237 swings before the pendulum with length 1.990 m has swung one more swing than the pendulum with length 2.000 m.