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A dust particle having mass equal to 10^-11g and velocity 10^-4 cm/s. calculate uncertainty in its position?
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A dust particle having mass equal to 10^-11g and velocity 10^-4 cm/s. ...
Uncertainty in Position of a Dust Particle
Position of a particle is uncertain due to its velocity and mass. To calculate the uncertainty in the position of a dust particle, we can use the Heisenberg Uncertainty Principle, which states that the product of the uncertainties in position and momentum of a particle is always greater than or equal to a certain constant.
Given Parameters:
- Mass (m) = 10^-11 g
- Velocity (v) = 10^-4 cm/s
Heisenberg Uncertainty Principle:
Δx * Δp ≥ h/4π
Where:
- Δx is the uncertainty in position
- Δp is the uncertainty in momentum
- h is the reduced Planck constant (h/2π)
Calculating Uncertainty in Momentum:
Momentum (p) = mass * velocity
Δp = mass * Δv + velocity * Δm
Δp = 10^-11 * Δ(10^-4) + 10^-4 * Δ(10^-11)
Δp = 10^-15 + 10^-15
Δp = 2 * 10^-15 g*cm/s
Using Uncertainty Principle to find Uncertainty in Position:
Δx * 2 * 10^-15 ≥ h/4π
Δx ≥ h/(2 * 10^-15 * 4π)
Δx ≥ 1.05 * 10^-10 cm
Therefore, the uncertainty in the position of the dust particle is approximately 1.05 * 10^-10 cm. This uncertainty arises due to the particle's mass and velocity, as described by the Heisenberg Uncertainty Principle.
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