AHH this problem is driving me nuts!
A small container of water is placed on a carousel inside a microwave oven, at a radius of 13.5 cm from the center. The turntable rotates steadily, turning through one revolution each 6.50 s. What angle does the water surface make with the horizontal?
I got an answer of 0.76212865 deg. but the online hw thing says that I'm within 10% of the correct answer. Please help...THANK YOU!!
I figured it out. I was using velocity instead of acceleration based off of a=v^2/r....
To find the angle that the water surface makes with the horizontal, we can use the concept of centrifugal force.
The centrifugal force is directed away from the center of rotation and can be calculated as:
F_c = m * r * ω²
where F_c is the centrifugal force, m is the mass of the water, r is the radius, and ω (omega) is the angular velocity.
Now, let's break this problem down step by step:
Step 1: Find the angular velocity (ω).
Given that the carousel completes one revolution in 6.50 seconds, we can calculate the angular velocity using the formula:
ω = 2π / T
where T is the time for one revolution.
ω = 2π / 6.50 s
Step 2: Calculate the centrifugal force (F_c).
The problem doesn't provide the mass of the water, so we can assume a reasonable value. Let's say the mass is 100 grams (0.1 kg).
F_c = m * r * ω²
F_c = 0.1 kg * 0.135 m * (ω)²
Step 3: Find the angle (θ) by using the gravitational force as the vertical component.
The gravitational force acts downward, and we can use it as the vertical component to find the angle. The gravitational force is given by:
F_g = m * g
where g is the acceleration due to gravity (approximated as 9.8 m/s²).
Equating F_c and F_g, we'll have:
m * r * ω² = m * g
Simplifying, we find:
r * ω² = g
Hence, ω² = g / r
Now we can substitute the value of ω² in the formula for F_c:
F_c = 0.1 kg * 0.135 m * (g / r)
Step 4: Calculate the angle (θ).
The angle (θ) is given by the equation:
tan(θ) = F_c / F_g
tan(θ) = (0.1 kg * 0.135 m * (g / r)) / (0.1 kg * g)
tan(θ) = 0.135 m / r
Now, we can calculate the angle θ:
θ = atan(0.135 m / r)
Plugging in the values:
θ = atan(0.135 m / 0.135 m)
θ = atan(1)
θ ≈ 45°
Therefore, the correct answer for the angle that the water surface makes with the horizontal is approximately 45 degrees.