1.1. To consider speed & velocity around a circleTo consider speed & velocity around a circle

2.2. To consider acceleration as a change in To consider acceleration as a change in velocityvelocity

3.3. To define an equation for centripetal To define an equation for centripetal accelerationacceleration

4.4. To define an equation for centripetal forceTo define an equation for centripetal force

Book Reference : Pages 24-25Book Reference : Pages 24-25

If an object is moving in a circle If an object is moving in a circle with a constant speed, it’s with a constant speed, it’s velocity is constantly changing....velocity is constantly changing....

Because the direction is Because the direction is constantly changing....constantly changing....

If the velocity is constantly If the velocity is constantly changing then by definition the changing then by definition the object is acceleratingobject is accelerating

If the object is accelerating, then If the object is accelerating, then an unbalanced force must existan unbalanced force must exist

Velocity v

acceleration

Velocity vB

Velocity vA

Consider an object moving Consider an object moving in circular motion with a in circular motion with a speed v which moves from speed v which moves from point A to point B in point A to point B in t t secondsseconds

(From speed=distance / time), the (From speed=distance / time), the distance moved along the arc AB, distance moved along the arc AB, s is vs is vt t

Velocity vB

Velocity vA

v

C A

B

The vector diagram shows The vector diagram shows the change in velocity the change in velocity v :v :

(v(vBB – v – vAA) )

Velocity vB

Velocity vA

The triangles ABC & the The triangles ABC & the vector diagram are similarvector diagram are similar

If If is small, then is small, then v / v = v / v = s / rs / r

Velocity vB

Velocity vA

v

C A

B

Substituting for Substituting for s = vs = vtt

v / v = vv / v = vt / rt / r

(a = change in velocity / time)(a = change in velocity / time)

a = a = v / v / t = vt = v2 2 / r/ r

We can substitute for angular velocity....We can substitute for angular velocity....

a = va = v2 2 / r/ r

From the last lesson we saw that:From the last lesson we saw that:

v = rv = r (substituting for v into above) (substituting for v into above)

a = (ra = (r))2 2 / r/ r

a = ra = r22

In exactly the same way as we can connect force In exactly the same way as we can connect force f f and acceleration and acceleration aa using Newton’s 2 using Newton’s 2ndnd law of law of motion, we can arrive at the centripetal force motion, we can arrive at the centripetal force which is keeping the object moving in a circlewhich is keeping the object moving in a circle

f = mvf = mv2 2 / r/ r

oror

f = mrf = mr22

Any object moving in a circle is acted upon by a Any object moving in a circle is acted upon by a single resultant force towards the centre of the single resultant force towards the centre of the circle. We call this the centripetal forcecircle. We call this the centripetal force

Gravity which keeps satellites in orbit around Gravity which keeps satellites in orbit around Earth and the Earth in orbit around the sun is a Earth and the Earth in orbit around the sun is a classic example of a centripetal force.classic example of a centripetal force.

Planet

satellite

Gravity

The wheel of the London Eye has a diameter of The wheel of the London Eye has a diameter of 130m and takes 30mins for 1 revolution. 130m and takes 30mins for 1 revolution. Calculate:Calculate:

a.a. The speed of the capsuleThe speed of the capsule

b.b. The centripetal accelerationThe centripetal acceleration

c.c. The centripetal force on a person with a The centripetal force on a person with a mass of 65kgmass of 65kg

The speed of the capsule :The speed of the capsule :

Using v = rUsing v = r

we know that we do a full revolution (2we know that we do a full revolution (2 rad) rad) in 30mins (1800s)in 30mins (1800s)

v = (130/2) x (2v = (130/2) x (2 / 1800) / 1800)

v = 0.23 msv = 0.23 ms-1-1

The centripetal acceleration:The centripetal acceleration:

Using a = vUsing a = v2 2 / r/ r

a = (0.23)a = (0.23)22 / (130/2) / (130/2)

a = 7.92 x 10a = 7.92 x 10-4-4 ms ms-2-2

The centripetal force:The centripetal force:

Using f = maUsing f = ma

F = 65 x 7.92 x 10F = 65 x 7.92 x 10-4-4

F = 0.051 NF = 0.051 N

An object of mass 0.15kg moves around a circular An object of mass 0.15kg moves around a circular path which has a radius of 0.42m once every 5s at path which has a radius of 0.42m once every 5s at a steady rate. Calculate:a steady rate. Calculate:

a.a. The speed and acceleration of the objectThe speed and acceleration of the object

b.b. The centripetal force on the objectThe centripetal force on the object

[.528 ms[.528 ms-1-1, 0.663ms, 0.663ms-2-2, 0.100N], 0.100N]