Circular Motion and Other Applications of Newton’s Laws Chapter 6 Circular Motion and Other Applications of Newton’s Laws

Circular Motion Objects traveling in circular paths Motion observed from an accelerating frame of reference Motion of an object through a viscous medium Introduction

Uniform Circular Motion, Acceleration A particle moves with a constant speed in a circular path of radius r with an acceleration. The magnitude of the acceleration is given by The centripetal acceleration, , is directed toward the center of the circle. The centripetal acceleration is always perpendicular to the velocity. Section 6.1

Uniform Circular Motion, Force A force, , is associated with the centripetal acceleration. The force is also directed toward the center of the circle. Applying Newton’s Second Law along the radial direction gives Section 6.1

Uniform Circular Motion, cont. A force causing a centripetal acceleration acts toward the center of the circle. It causes a change in the direction of the velocity vector. If the force vanishes, the object would move in a straight-line path tangent to the circle. Section 6.1

Conical Pendulum The object is in equilibrium in the vertical direction . It undergoes uniform circular motion in the horizontal direction. ∑Fy = 0 → T cos θ = mg ∑Fx = T sin θ = m ac v is independent of m Section 6.1

Motion in a Horizontal Circle The speed at which the object moves depends on the mass of the object and the tension in the cord. The centripetal force is supplied by the tension. The maximum speed corresponds to the maximum tension the string can withstand. Section 6.1

Horizontal (Flat) Curve The car as a particle in uniform circular motion in the horizontal direction. The car as a particle in equilibrium in the vertical direction. The force of static friction supplies the centripetal force. The maximum speed at which the car can negotiate the curve is: Note, this does not depend on the mass of the car. Section 6.1

Banked Curve These are designed with friction equaling zero. The car as a particle in equilibrium in the vertical direction. The car as a particle in uniform circular motion in the horizontal direction. There is a component of the normal force that supplies the centripetal force. The angle of bank is found from Section 6.1

Banked Curve, 2 The angle is independent of the mass of the vehicle. If the car rounds the curve at less than the design speed, friction is necessary to keep it from sliding down the bank. If the car rounds the curve at more than the design speed, friction is necessary to keep it from sliding up the bank. Section 6.1

Non-Uniform Circular Motion The acceleration and force have tangential components. produces the centripetal acceleration produces the tangential acceleration The total force is Section 6.2

Vertical Circle with Non-Uniform Speed The gravitational force exerts a tangential force on the object. Look at the components of Fg Model the sphere as a particle under a net force and moving in a circular path. Not uniform circular motion The tension at any point can be found. Section 6.2

Top and Bottom of Circle The tension at the bottom is a maximum. The tension at the top is a minimum. If Ttop = 0, then Section 6.2

Motion in Accelerated Frames A fictitious force results from an accelerated frame of reference. The fictitious force is due to observations made in an accelerated frame. A fictitious force appears to act on an object in the same way as a real force, but you cannot identify a second object for the fictitious force. Remember that real forces are always interactions between two objects. Simple fictitious forces appear to act in the direction opposite that of the acceleration of the non-inertial frame. Section 6.3

“Centrifugal” Force From the frame of the passenger (b), a force appears to push her toward the door. From the frame of the Earth, the car applies a leftward force on the passenger. The outward force is often called a centrifugal force. It is a fictitious force due to the centripetal acceleration associated with the car’s change in direction. In actuality, friction supplies the force to allow the passenger to move with the car. If the frictional force is not large enough, the passenger continues on her initial path according to Newton’s First Law. Section 6.3

Fictitious Forces, examples Although fictitious forces are not real forces, they can have real effects. Examples: Objects in the car do slide You feel pushed to the outside of a rotating platform The force is responsible for the rotation of weather systems, including hurricanes, and ocean currents. Section 6.3

Fictitious Forces in Linear Systems The inertial observer the sphere as a particle under a net force in the horizontal direction and a particle in equilibrium in the vertical direction. The non-inertial observer the sphere as a particle in equilibrium in both directions. The inertial observer (a) at rest sees The non-inertial observer (b) sees These are equivalent if Ffictiitous = ma Section 6.3

Example 1

Example 2

Example 3