A simple pendulum consists of a ball connected to one end of a thin brass wire. The period of the pendulum is 2.0000 s. The temperature rises by 149° C, and the length of the wire increases. Determine the period of the heated pendulum
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The period of the heated pendulum is 2.0014 s.
To determine the period of the heated pendulum, we need to consider the effect of the change in length due to the temperature rise. The period of a simple pendulum is given by the formula:
T = 2π√(L/g)
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity (approximately 9.8 m/s²).
Since the length of the wire increases due to the temperature rise, we need to determine the new length of the pendulum. The thermal expansion of brass can be described by the linear expansion coefficient, which tells us how much the length of the wire will increase for a given temperature change.
To find the new length of the wire, we can use the formula:
ΔL = αL₀ΔT
where ΔL is the change in length, α is the linear expansion coefficient of brass, L₀ is the initial length of the wire, and ΔT is the change in temperature.
However, we do not have the values for α and L₀, so we can't directly calculate the change in length. In this case, we can assume that the change in length is small compared to the initial length, and we can use a linear approximation:
ΔL ≈ L₀αΔT
Using this approximation, we can compute the new length of the pendulum:
L' = L₀ + ΔL
Now that we have the new length, we can substitute it into the period formula to obtain the period of the heated pendulum:
T' = 2π√(L'/g)
Substituting the given values and calculating:
T' = 2π√[(L₀ + L₀αΔT)/g]
Finally, we can simplify the equation and calculate the period of the heated pendulum:
T' = 2π√(L₀/g) * √(1 + αΔT)
where √(1 + αΔT) represents the correction factor due to the temperature rise.
To determine the period of the heated pendulum, we need to take into account the change in length caused by the increase in temperature.
The formula to calculate the period of a simple pendulum is:
T = 2π * √(L / g)
where:
T = period of the pendulum
L = length of the pendulum
g = acceleration due to gravity (approximately 9.8 m/s²)
Since the length of the wire increases due to the rise in temperature, we can express the new length as:
L_new = L_initial(1 + αΔT)
where:
L_new = new length of the wire
L_initial = initial length of the wire
α = coefficient of linear expansion of the material (for brass, α = 0.000019/°C)
ΔT = change in temperature in Celsius
In this case, the period of the pendulum before the temperature increase is 2.0000 seconds, and the change in temperature is 149°C.
Step 1: Convert the initial period to angular frequency (ω)
ω = 2π / T
ω = 2π / 2.0000
ω ≈ 3.1416 rad/s
Step 2: Calculate the new length of the wire
L_new = L_initial * (1 + αΔT)
Let's assume the initial length of the wire is L_initial = L.
L_new = L * (1 + αΔT)
L_new ≈ L * (1 + 0.000019/°C * 149°C)
L_new ≈ L * (1 + 0.002831 °C)
L_new ≈ L * 1.002831
Step 3: Calculate the new period of the pendulum
T_new = 2π * √(L_new / g)
T_new = 2π * √(L * 1.002831 / g)
T_new ≈ 2π * √(L / g) * √1.002831
T_new ≈ T * √1.002831
Step 4: Calculate the new period in seconds
T_new ≈ 2.0000 s * √1.002831
T_new ≈ 2.0000 s * 1.001413
Therefore, the period of the heated pendulum is approximately 2.0028 seconds.