Dislocation e velocitate

Dislocation

• Linear dislocation (s) is the length of a curve formed when a body moves in a circle around a point.
• Angular dislocation (θ) is the angle subtended by a body moving in a circle around a point.
• Angular dislocation is measured in radians.
• One radian is defined as the angular dislocation when the linear dislocation is equal to the radius of the track.
• To convert radians to degrees, multiply by 180/π
• To convert degrees to radians, multiply by π/180

Velocity

• Linear velocity (v) is the rate of change of linear dislocation around a point.
• Linear velocity has SI units of m/s
• The direction of v is tangent to the point on the circular track.
• Angular velocity (ω) is the rate of change of angular dislocation around a point.
• Angular velocity has SI units of rad/s.
• ω = 2π/T
• ω = 2πf
• N(rpm) means cycles per minute.
• N = f x 60
• ω = 2πN/60
• Δs = rΔθ
• Dividing it by Δt, Δs/Δt = rΔθ/Δt
• v = rω
• With ω constant, v ∝ r
• In the same time interval, more dislocation is covered by a point P
• Centripetal acceleration is the rate of change of linear velocity about a point due to the change in the direction of motion.
• a_c = vω
• a_c = v x v/r
• a_c = v²/r
• a_c = rω²
• The direction of centripetal acceleration is towards the center of rotation.

Centripetal Force

• Centripetal force is the resultant force that tends to rotate an object in a circular path about a point.
• F_resultant = ma_c
• F_c = mrω²
• F_c = mv²/r
• F_c = mrω
• Centripetal force is a name given to any force that causes an object to rotate in a circle about a point.
• Examples of centripetal force include:
  • Stone and rope, where the centripetal force is tension in the rope
  • Planetary motion, where the centripetal force is gravitational force
  • A charged particle in a magnetic field, where the centripetal force is magnetic force
  • A car moving around a turn, where the centripetal force is friction
  • A bucket full of water, where the centripetal force is the reaction from the bucket

Stone in Horizontal Circle

• F_c = Tsinθ
• Tsinθ = mrω²
• Tcosθ = W

Stone Moving in a Vertical Circle

• A stone is moving at a constant angular velocity in a fixed circle.
• F_c is a constant.
• W + F_c = T (maximum tension)
• T = F_c
• W + T = F_c; hence, T = F_c - W
  • If F_c is smaller than W, the stone will drop.
  • If F_c = W, there will be no tension in the string as the stone crosses the vertical position.
• If F_c > W, the tension will also act to help balance F_c.
• Wcosθ + F_c = T

Further Examples

• Friction force acts as a centripetal force
• Rsinθ + Fcosθ
• For minimum angular velocity to keep water in a bucket, R = 0
• Centrifugal force must be enough to counter the weight of water.
• F_c = W + 0
• mrω² = mg
• ω² = g/r
• ω = √(g/r)