A clock has a copper pendulum with a period of 1.000 s at 17.3°C. Suppose the clock is moved to a location where the average temperature is 33.9°C.
Determine the new period of the clock's pendulum.
To determine the new period of the clock's pendulum, we can use the formula for the period of a simple pendulum:
T = 2π√(L/g)
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
Since the length of the pendulum remains constant, we can assume that L is the same at both temperatures.
However, the acceleration due to gravity can vary due to the change in temperature. The acceleration due to gravity, g, can be approximated as:
g = g₀(1 + αΔT)
where g₀ is the acceleration due to gravity at a reference temperature, α is the linear coefficient of expansion for the material, and ΔT is the change in temperature.
In this case, the pendulum is made of copper, which has a linear coefficient of expansion of α = 0.000016/°C, and the reference temperature is 17.3°C.
To find the new period, we need to calculate the change in temperature, ΔT, and the new acceleration due to gravity, g.
ΔT = 33.9°C - 17.3°C = 16.6°C
g = 9.81 m/s² * (1 + (0.000016/°C * 16.6°C))
Now, we can plug the value of g into the formula for the period of the pendulum to find the new period, T_new.
T_new = 2π√(L/g)
(T_new = 2π√(L/ (9.81 m/s² * (1 + (0.000016/°C * 16.6°C)) ))
After evaluating this equation, we will find the new period of the clock's pendulum.
To determine the new period of the clock's pendulum at the new temperature, you need to use the formula for calculating the period of a pendulum. The formula is given by:
T = 2π√(L/g)
Where T is the period of the pendulum, π is a mathematical constant (approximately equal to 3.14159), L is the length of the pendulum, and g is the acceleration due to gravity.
Since we are given the period of the pendulum at 17.3°C, we can set up the equation as follows:
T₁ = 2π√(L/g)
To find the period of the pendulum at 33.9°C, we need to find the new length of the pendulum. The length of a pendulum is affected by temperature, as the coefficient of linear expansion for copper is not zero.
The formula for calculating the change in length due to temperature change is given by:
ΔL = (α)(L₀)(ΔT)
Where ΔL is the change in length, α is the coefficient of linear expansion for copper, L₀ is the original length of the pendulum, and ΔT is the change in temperature.
Assuming the length of the pendulum remains constant as it is moved to the new location, we can calculate the change in length as follows:
ΔL = (α)(L₀)(ΔT)
Substituting the given values:
ΔL = (α)(L₀)(T₂ - T₁)
Where T₂ is the new average temperature and T₁ is the initial average temperature.
Finally, we can calculate the new period of the pendulum at the new temperature (T₂) using the following equation:
T₂ = T₁ + ΔT
Substituting the values, we have:
T₂ = T₁ + ΔL
Now you can input the values of the original temperature (T₁), the new temperature (T₂), and the length of the pendulum (L₀) into the equations mentioned above to find the new period of the clock's pendulum. Remember to use the appropriate units for temperature and length.