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Centripetal acceleration & Force

Centripetal acceleration & Force. Learning Objectives. Book Reference : Pages 24-25. Centripetal acceleration & Force. To consider speed & velocity around a circle To consider acceleration as a change in velocity To define an equation for centripetal acceleration

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Centripetal acceleration & Force

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  1. Centripetal acceleration & Force

  2. Learning Objectives Book Reference : Pages 24-25 Centripetal acceleration & Force To consider speed & velocity around a circle To consider acceleration as a change in velocity To define an equation for centripetal acceleration To define an equation for centripetal force

  3. If an object is moving in a circle with a constant speed, it’s velocity is constantly changing.... Because the direction is constantly changing.... If the velocity is constantly changing then by definition the object is accelerating If the object is accelerating, then an unbalanced force must exist Velocity v Centripetal Acceleration : Introduction acceleration

  4. Consider an object moving in circular motion with a speed v which moves from point A to point B in t seconds Velocity vB Centripetal Acceleration : Proof 1 B Velocity vA  C A (From speed=distance / time), the distance moved along the arc AB, s is vt v The vector diagram shows the change in velocity v : (vB – vA) Velocity vA  Velocity vB

  5. The triangles ABC & the vector diagram are similar Velocity vB Centripetal Acceleration : Proof 2 B Velocity vA  C A If  is small, then v / v = s / r Substituting for s = vt v / v = vt / r (a = change in velocity / time) a = v / t = v2 / r v Velocity vA  Velocity vB

  6. We can substitute for angular velocity.... • a = v2 / r • From the last lesson we saw that: • v = r (substituting for v into above) • a = (r)2 / r • a = r2 Centripetal Acceleration : angular

  7. In exactly the same way as we can connect force f and acceleration a using Newton’s 2nd law of motion, we can arrive at the centripetal force which is keeping the object moving in a circle • f = mv2 / r • or • f = mr2 • Any object moving in a circle is acted upon by a single resultant force towards the centre of the circle. We call this the centripetal force Centripetal Force

  8. Gravity which keeps satellites in orbit around Earth and the Earth in orbit around the sun is a classic example of a centripetal force. Centripetal Force : Gravity & Orbits satellite Gravity Planet

  9. The wheel of the London Eye has a diameter of 130m and takes 30mins for 1 revolution. Calculate: • The speed of the capsule • The centripetal acceleration • The centripetal force on a person with a mass of 65kg Worked example 1

  10. The speed of the capsule : • Using v = r • we know that we do a full revolution (2 rad) in 30mins (1800s) • v = (130/2) x (2 / 1800) • v = 0.23 ms-1 Worked example 1

  11. The centripetal acceleration: • Using a = v2 / r • a = (0.23)2 / (130/2) • a = 7.92 x 10-4 ms-2 • The centripetal force: • Using f = ma • F = 65 x 7.92 x 10-4 • F = 0.051 N Worked example 1

  12. An object of mass 0.15kg moves around a circular path which has a radius of 0.42m once every 5s at a steady rate. Calculate: • The speed and acceleration of the object • The centripetal force on the object • [.528 ms-1, 0.663ms-2, 0.100N] Problem 1

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