A standard DVD has a diameter of 12.0 cm, and a hole that is centered on the axis of symmetry and has a diameter of 1.50 cm. The DVD’s thickness is 1.2 mm, and it's mass is 17 g. If a (very thin) pin extending through the hole of the DVD suspends it in a vertical orientation, as shown in the figure, the DVD may oscillate about an axis parallel to the pin, rocking back and forth. How long does it take to complete 9 complete oscillations?
Well, well, well, looks like the DVD wants to rock and roll! Let's calculate how long it takes to complete 9 complete oscillations, shall we?
First things first, we need to find the moment of inertia of the DVD. The moment of inertia of a thin disc about its axis of symmetry is given by the formula:
I = (1/4) * m * (R^2 + r^2)
Where:
m is the mass of the DVD (17 g converted to kg)
R is the outer radius (half the diameter of the DVD)
r is the inner radius (half the diameter of the hole)
Let's do the math:
m = 17 g = 0.017 kg
R = (12.0 cm / 2) = 6.0 cm = 0.06 m
r = (1.50 cm / 2) = 0.75 cm = 0.0075 m
I = (1/4) * 0.017 kg * ((0.06 m)^2 + (0.0075 m)^2)
I = 0.0000704175 kgm²
Now, to calculate the oscillation time period (T) of a physical pendulum, we use the formula:
T = 2π * √(I / (m * g * d))
Where:
π is Pi...yum!
I is the moment of inertia we just calculated
m is the mass of the DVD again
g is the acceleration due to gravity (9.8 m/s²)
d is the distance from the pivot point to the center of mass of the DVD
But wait, we need to find 'd' first. The center of mass of a uniform disc is located at a distance equal to half its thickness (t) from the axis of symmetry. In this case, t = 1.2 mm = 0.0012 m.
d = (0.06 m + 0.0075 m) / 2 + 0.0012 m
d = 0.0349 m
Now, let's put all the pieces together and calculate the oscillation time period:
T = 2π * √(0.0000704175 kgm² / (0.017 kg * 9.8 m/s² * 0.0349 m))
T = 2π * √(0.0026774907)
T = 2π * 0.05178397311
T ≈ 0.325 s
And there you have it! It takes approximately 0.325 seconds for the DVD to complete 9 oscillations. Rock on, DVD!
To find the time it takes to complete 9 complete oscillations, we need to use the formula for the period of oscillation. The period (T) is defined as the time it takes for one complete cycle.
The period of oscillation (T) can be calculated using the formula:
T = 2π√(I/mg)
Where:
T = period of oscillation
π = pi (approximately 3.14159)
I = moment of inertia of the DVD
m = mass of the DVD
g = acceleration due to gravity (approximately 9.8 m/s²)
The moment of inertia (I) for a thin disc oscillating about an axis perpendicular to its plane can be calculated using the formula:
I = (1/4) * m * (r²1 + r²2)
Where:
m = mass of the DVD
r₁ = radius of the disc (outer radius)
r₂ = radius of the hole (inner radius)
Given information:
diameter of the DVD (r₁) = 12.0 cm = 0.12 m
diameter of the hole (r₂) = 1.50 cm = 0.015 m
thickness of the DVD = 1.2 mm = 0.0012 m
mass of the DVD (m) = 17 g = 0.017 kg
acceleration due to gravity (g) = 9.8 m/s²
First, let's calculate the radius of the disc (r₁) by halving its diameter:
r₁ = 0.12 m / 2 = 0.06 m
Next, let's calculate the moment of inertia (I) using the formula:
I = (1/4) * m * (r₁² + r₂²)
I = (0.25) * 0.017 kg * (0.06 m)² + (0.015 m)²)
I = 5.1 x 10^-5 kg m²
Now, let's substitute the values of I, m, and g into the formula for the period of oscillation:
T = 2π√(I/mg)
T = 2π√((5.1 x 10^-5 kg m²) / (0.017 kg * 9.8 m/s²))
T = 1.28 s
Finally, to find the time it takes to complete 9 complete oscillations, we multiply the period (T) by the number of oscillations (9):
Time = T * number of oscillations = 1.28 s * 9 = 11.52 s
Therefore, it takes approximately 11.52 seconds to complete 9 complete oscillations.
To find the time it takes to complete 9 complete oscillations, we need to use the formula for the period of oscillation.
The period of oscillation, T, is the time it takes for one complete cycle. It is inversely proportional to the square root of the rotational inertia, I, of the system.
For a thin disk like the DVD, the rotational inertia can be calculated using the formula:
I = (1/2) * m * r^2
where m is the mass of the DVD and r is the radius of the DVD.
To find the radius, we can subtract half the diameter of the hole from half the diameter of the DVD:
r = (12.0 cm / 2 - 1.50 cm / 2)
Next, we calculate the rotational inertia:
I = (1/2) * (17 g) * (r^2)
Now, we can calculate the period of oscillation:
T = 2π * √(I / (m * g * h))
where g is the acceleration due to gravity and h is the height from the center of rotation to the center of mass.
In this case, since the DVD is hanging vertically, the height h is equal to the thickness of the DVD:
h = 1.2 mm
Now, we can substitute the values and solve for T:
T = 2π * √((1/2) * (17 g) * (r^2) / (m * g * h))
To convert the thickness and diameter from centimeters to meters, and the mass from grams to kilograms, we divide by 100:
T = 2π * √((1/2) * (17 g / 1000 kg) * ((r / 100)^2) / ((m / 1000 kg) * 9.8 m/s^2 * (h / 100 m)))
Now, we can calculate the time it takes to complete 9 complete oscillations by multiplying the period by 9:
9T = 9 * 2π * √((1/2) * (17 g / 1000 kg) * ((r / 100)^2) / ((m / 1000 kg) * 9.8 m/s^2 * (h / 100 m)))
Finally, we can calculate the value using a calculator or by plugging in the values and evaluating the expression.