Torque (Moment Force)

• The effectiveness of force in producing rotation about that axis.

• It is measured by the product of the force and the perpendicular distance from the axis of rotation to the line of action of force.

Mathematical Definition of torque

• Magnitude:r F

•Direction is determined by the right hand rule.

r FNote: for fixed rotational axis motion, only

components of and in the x-y plane will contribute to the torque along the rotational axis

Practical Application

• Which of the following forces provide the highest torque from O and in what direction.

Example 6  The Achilles Tendon

Figure (a) shows the ankle joint and the Achilles tendon attached to the heel at point P. The tendon exerts a force of magnitude F=720 N, as Figure (b) indicates. Determine the torque (magnitude and direction) of this force about the ankle joint, which is located 3.6×10–2 m away from point P.

Newton's Second Law for Rotation

Newton's Second Law for Rotation

ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS:

netz zI

Requirement: a must be expressed in rad/s2.

Important: and are vector components along the same axis

Example 7  The Torque of an Electric Saw Motor

1. The motor in an electric saw brings the circular blade from rest up to the rated angular velocity of 80.0 rev/s in 240.0 rev. One type of blade has a moment of inertia of 1.41×10–2 kg · m2. What net torque (assumed constant) must the motor apply to the blade?

Example 8  2. Figure a shows a uniform disk, with

mass M = 2.5 kg and radius R = 20 cm, mounted on a fixed horizontal axle. A block with mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk. Find the acceleration of the falling block, the angular acceleration of the disk, and the tension in the cord. The cord does not slip, and there is no friction at the axle.

    

Rotational Work

The rotational work WR done by a constant torque in turning an object through an angle θ is:

Requirement: θ must be expressed in radians. Unit of work: joule (J)

For variable torque, rotational work is:

Rotational Net Work-Kinetic Energy Theorem

• 1. A ball of mass M and radius R starts from rest at a height of 2.00m and rolls down a 30.0° slope. What is the linear speed of the ball when it leaves the incline. Assume that the ball rolls without slipping.

Work and Energy in Rotational Motion

Corresponding Relations for Translational and Rotational Motion

• #59, 61 p. 270