An aluminum wire of radius 3.10 10-4 m is stretched between the ends of 2 concrete blocks. When the system (wire and concrete) is at 35°C, the tension in the wire is 57.0 N. What is the tension in the wire when the system is heated to 185°C?
To find the tension in the wire when the system is heated to 185°C, we can use the formula for thermal expansion:
ΔL = α * L * ΔT
where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature.
First, let's find the change in length of the wire due to the change in temperature. We can use the formula:
ΔL = α * L * ΔT
where α is the coefficient of linear expansion for aluminum, and ΔT is the change in temperature.
Next, calculate the change in length:
ΔL = α * L * ΔT
ΔL = (23 * 10^-6 / °C) * (2πr) * ΔT
ΔL = (23 * 10^-6 / °C) * (2π * 3.10 * 10^-4 m) * (185 - 35)°C
ΔL = (23 * 10^-6 / °C) * (2π * 3.10 * 10^-4 m) * 150°C
ΔL = (23 * 10^-6 / °C) * (2π * 3.10 * 10^-4 m) * 150°C
ΔL = (23 * 10^-6 / °C) * (2π * 3.10 * 10^-4 m) * 150°C
ΔL = 0.0216 m (rounded to 4 decimal places)
The total length of the wire, L, is given by:
L = 2πr
L = 2π * 3.10 * 10^-4 m
L = 1.94 * 10^-3 m (rounded to 4 decimal places)
Now, let's calculate the new length of the wire:
New length = Original length + Change in length
New length = L + ΔL
New length = 1.94 * 10^-3 m + 0.0216 m
New length = 0.0236 m (rounded to 4 decimal places)
Finally, we can calculate the tension in the wire when the system is heated to 185°C. The tension in the wire can be calculated using Hooke's Law:
Tension = Young's Modulus * (Change in length / Original length)
The Young's modulus for aluminum is typically around 70 GPa (70 * 10^9 Pa).
Tension = (70 * 10^9 Pa) * (0.0216 m / 1.94 * 10^-3 m)
Tension = 778 N (rounded to 3 significant figures)
Therefore, the tension in the wire when the system is heated to 185°C is approximately 778 N.
To determine the tension in the wire when the system is heated to 185°C, we need to consider the thermal expansion of the aluminum wire. The formula to calculate the change in length due to thermal expansion is given by:
ΔL = α * L * ΔT
where:
ΔL is the change in length of the wire
α is the coefficient of linear expansion of aluminum (which is 0.000022/°C)
L is the original length of the wire
ΔT is the change in temperature of the system (185°C - 35°C)
Since the wire is stretched between the concrete blocks, the change in length of the wire will cause a change in tension. The change in tension is given by:
ΔTension = Tension_initial * (ΔL / L)
where:
ΔTension is the change in tension in the wire
Tension_initial is the initial tension in the wire (57.0 N)
ΔL is the change in length of the wire
L is the original length of the wire
To find the tension in the wire when the system is heated to 185°C, we can first calculate the change in length of the wire using the formula mentioned earlier:
ΔL = (0.000022/°C) * L * (185°C - 35°C)
Now we can substitute the values into the equation for the change in tension:
ΔTension = 57.0 N * (ΔL / L)
Calculating this expression will give us the change in tension in the wire. To find the final tension in the wire, we subtract the change in tension from the initial tension:
Tension_final = Tension_initial - ΔTension
Substituting the known values will give us the final tension in the wire when the system is heated to 185°C.