Respuesta :
Answer:
The angular speed is 7.01 rad/s
The loss of energy to thermal energy is 965.6 J
Explanation:
Conservation of angular momentum says that 'the angular momentum must be the same before and after the clump of clay is attached to the turntable'.
The angular momentum is defined as: L = I*ω
with I = the moment of inertia
with w= the angular speed
The angular momentum before must be equal to the angular momentum after.
Angular momentum before: Lb = It*ω0
Angular momentum after: La = (It + Ic)*ω1
So Lb = La⇔ It*ω0 = (It + Ic)*ω1
with ω1 = the angular speed after (ω1 = ω0*[It/(It + Ic)])
ω₁ = I₀ω₀ / I₁
I₀ = The moment of inertia of the turntable
= The moment of inertia of a cylinder with mass m and radius r around its z axis = ½mr²., and can be calculated as followed:
I₀ = ½(28)(3.3)² = 152.46 kgm²
I₁ = I₀ + I(clay) = 152.46kgm² + m(clay)*r(clay)² = 152.46 kgm² + (10)(1.80)² = 184.86 kgm²
To calculate ω₁ we use the following formula:
ω₁ = I₀ω₀ / I₁ = 152.46*8.5/ 184.86 = 7.01 rad/s
Calculate the loss of energy to thermal energy
Energy is not conserved. Some amount of energy is lost when the clay deforms and sticks to the turntable! This change in energy is given by:
Et= E₁ - E₀ = (0.5)(184.86)(11.078)² - (0.5)(61.8)(4π)² = -577.94 J
with E₁= ½I₁ω₁² = ½ * 184.86 * 7.01² = 4542.02
with E₀ = ½I₀ω₀² = ½ * 152.46 *8.5² = 5507.62
Et= E₁ - E₀ = 4542.02 - 5507.62 = - 965.6 J
This value is negative, this means there is 965.6 J energy lost to thermal energy
- The angular speed of the clay is 7.01 rad/s
- The loss of energy to thermal energy is 965.6 J
What is Angular momentum?
This is defined as the property of any rotating object given by moment of inertia times angular velocity.
Conservation of angular momentum says that the angular momentum is constant except an external torque is applied.
Angular momentum (L) = I*ω
where I is the moment of inertia, w is the angular speed
Angular momentum before: Lb = It*ω0
Angular momentum after: La = (It + Ic)*ω1
It*ω0 = (It + Ic)*ω1
ω1 = the angular speed after
ω1 = ω0*[It/(It + Ic)])
ω₁ = I₀ω₀ / I₁
I₀ = The moment of inertia of the turntable
The moment of inertia of a cylinder = ½mr²
I₀ = ½(28)(3.3)² = 152.46 kgm²
I₁ = I₀ + I(clay) = 152.46kgm² + m(clay)*r(clay)² = 152.46 kgm² + (10)(1.80)² = 184.86 kgm²
ω₁ = I₀ω₀ / I₁ = 152.46*8.5/ 184.86 = 7.01 rad/s
- Loss of energy to thermal energy
Energy is not conserved as some amount of energy is lost when the clay deforms and sticks to the turntable. We can deduce that:
Et= E₁ - E₀ = (0.5)(184.86)(11.078)² - (0.5)(61.8)(4π)² = -577.94 J
E₁= ½I₁ω₁² = ½ × 184.86 × 7.01² = 4542.02
E₀ = ½I₀ω₀² = ½ × 152.46 × 8.5² = 5507.62
Et= E₁ - E₀ = 4542.02 - 5507.62 = - 965.6 J
This value is negative which signifies 965.6 J being lost in the form of thermal energy.
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