R

v

v

Resultant force toward center Fc =

mv2

R

Consider TOP of circle:

AT TOP:

T

mg

T

mg

+

mg + T = mv2

R

T = - mg

mv2

R

R

v

v

Resultant force toward center

Fc = mv2

R

Consider bottom of circle:

AT Bottom:

T

mg

+

T - mg = mv2

R

T = + mg

mv2

R

T

mg

R

v

v

AT TOP:

T

mg+

AT BOTTOM:

T

mg

+

T = - mg mv2

R

T = + mg

mv2

R

R

v

v

T

mg

mg + T = mv2

RAt Top:

T = - mg mv2

R

T = 25 N - 19.6 N T = 5.40 NT = 5.40 N

22(2kg)(10m/s)

2 kg(9.8 m/s )8 m

T

A 2-kg rock swings in a vertical circle of radius 8 m. The speed of the rock as it passes its highest point is 10 m/s. What is tension T in rope?

R

v

vT

mg

T - mg = mv2

RAt Bottom:

T = + mg mv2

R

T = 25 N + 19.6 N T = 44.6 NT = 44.6 N

22(2kg)(10m/s)

2 kg(9.8 m/s )8 m

T

A 2-kg rock swings in a vertical circle of radius 8 m. The speed of the rock as it passes its lowest point is 10 m/s. What is tension T in rope?

Torque is defined as the tendency to produce a change in rotational motion.

Torque is defined as the tendency to produce a change in rotational motion.

Examples::

What is torque ?

What would help tighten the lug nuts, a short handle or a longer handle?

How do you produce more torque for picture (a) and picture (b)?

Torque is proportional to the magnitude of F and to the distance r from the axis. Thus, a tentative formula might be:

Torque is proportional to the magnitude of F and to the distance r from the axis. Thus, a tentative formula might be:

= Fr = Fr Units: Nm or lbft

6 cm

40 N

= (40 N)(0.60 m)

= 24.0 N= 24.0 Nmm

= 24.0 Nm = 24.0 Nm

• The magnitude of the applied force.• The direction of the applied force.• The location of the applied force.

• The magnitude of the applied force.• The direction of the applied force.• The location of the applied force.

20 N20 N

Magnitude of forceMagnitude of force

40 N40 N

The 40-N force The 40-N force produces twice the produces twice the torque as does the torque as does the

20-N force.20-N force.

Each of the 20-N Each of the 20-N forces has a different forces has a different

torque due to the torque due to the direction of force.direction of force. 20 N20 N

Direction of ForceDirection of Force

20 N20 N

20 N20 N20 N

Location of forceThe forces nearer the

end of the wrench have greater torques.

20 N20 N

Torque is a vector quantity that has direction as well as magnitude.

Torque is a vector quantity that has direction as well as magnitude.

Turning the handle of a screwdriver clockwise and then counterclockwise will advance the screw first inward and then

outward.

By convention, counterclockwise torques are positive and clockwise torques are

negative.

Positive torque: Counter-clockwise,

out of page

cw

ccw

Negative torque: clockwise, into page

• Extend line of action, draw, calculate r.

= (80 N)(0.104 m) = 8.31 N m = (80 N)(0.104 m) = 8.31 N mr = 12 cm sin 600 = 10.4 cm

r = 12 cm sin 600 = 10.4 cm

An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque.