Rotational Motion and Equilibrium Chapter 8 Rotational Motion and Equilibrium

Rigid Body An object or system of particles in which the distances between particles are fixed. Translational (sliding) and rotational (spinning) motion becomes relevant when we consider rigid bodies.

Translational vs Rotational Motion Translational Motion: Every particle in the rigid body has the same instantaneous velocity (no rotation). Rotational Motion: Every particle in the rigid body has the same angular velocity and travels in circles around a fixed axis.

Rotation and Translation

Rolling Motion Rolling without slipping is a combination of rotation and translation.

Torque A force is necessary to produce changes in rotational motion as well as translational. In pure translational motion, the translational acceleration is proportional to the net force. Fnet = ma In pure rotational motion, the angular acceleration is related to the net force AND to the perpendicular distance between the axis of rotation and the line of action. We call this distance the lever arm or r

Torque Terminology Line of Action – an imaginary line extending through the force vector Lever Arm – r - the perpendicular distance from the axis of rotation to the line of action

Line of Action and Lever Arm

Line of Action and Lever Arm

Torque Torque = (lever arm distance)·(Force) Torque = r ·F Torque = r F sinθ T = r F sin θ

Torque Direction The right hand rule can be used to find the direction of Torque as r X F T = r x F Alternatively, torques that cause counterclockwise rotation are taken to be positive and torques causing clockwise rotation are taken to be negative.

Torque T = r F sin θ Torque is a vector, r is a vector and F is a vector! Torque is measured in [m·N] The unit for Torque is NOT equivalent to a Joule [N·m] Right hand rule can be used to find the direction of torque.

Examples When you lift up on something with your forearm, torque is applied on the lower arm by the biceps muscle. With the axis of rotation through the elbow joint and the muscle attached 4.0 cm from the joint, what is the torque if the muscle exerts 600N of force?

Equilibrium Balanced forces create translational equilibrium. Balanced torques create rotational equilibrium. Mechanical equilibrium occurs when forces and torques balance. Static equilibrium occurs when a rigid body is at rest and forces and torques balance.

Example A picture hangs motionless on a wall. The picture has a mass of 3.0 kg. Find the magnitude of the tension in the wires if one wire makes a 50° angle and the other wire makes a 45° angle.

Example: Rotational Equilibrium Three masses are suspended from a meter stick balanced at the halfway point . 25g hangs from 0 cm and 75g hangs from 20cm. Unknown mass hangs at 85cm. Neglecting the mass of the meter stick, how much mass must be suspended on the right side for the system to be in static equilibrium?

Example A ladder with a mass of 15 kg rests against a smooth wall. A painter who has mass 78 kg stands on the ladder as shown. What is the magnitude of the frictional force that acts on the bottom to keep the ladder from sliding?