3) Uniform circular motion: to further understand acceleration in polar coordinates

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Elizabeth Gibson
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1 Physics 201 Lecture 7 Reading Chapter 5 1) Uniform circular motion: velocity in polar coordinates No radial velocity v = dr = dr Angular position: θ Angular velocity: ω Period: T = = " dθ dθ r + r θ = θ = rωθ = v θ When the motion is only in a circle v = v 2) Acceleration in polar coordinates = d v r r + d v θ = a r + a θ 3) Uniform circular motion: to further understand acceleration in polar coordinates For uniform circular motion (these calculation are for this specific case) v = dr = dr dθ dθ r + r θ = θ = rωθ = v θ (the term in multiplying r is zero because r is not changing) then " = rω dθ and specifically are more complex that you think " In an absolute sense the direction of θ changes as you go around the circle. find this acceleration. Consider this diagram: We want to

circular motion, is constant. Therefore the direction of a is r. Calculating the magnitude of the inward acceleration.

2 The average acceleration points roughly toward the center (along r) in the diagram. " We want to precisely understand the direction of a Direction: if a pointed in the θ direction it would increase or decrease the tangential velocity, which in the case we are considering, uniform circular motion, is constant. Therefore the direction of a is r. Calculating the magnitude of the inward acceleration. From the diagram above a v = lim t = lim v sin θ t v θ dθ = lim = v t = v ω = rω = v r (in this context v is sometimes called v ) then a = rω = v dθ = rω = rω r = v r = a r r a, the centripetal or center seeking acceleration, can be thought of as a central value of radial acceleration that will keep a particle with tangential velocity v moving in a circle. Add a positive acceleration and you will move outward in radius or and a negative acceleration and you will move inward. 4) Relative velocities Often we have to consider moving frames of reference. Examples: An airplane or bird moving within the moving frame of the air (wind).

3 A swimmer moving in the moving frame of a river. Taking the swimming case we might want to understand the movement both in the frame of the river (relative to the moving river) and the frame of the earth (relative to the shore). If a swimmer points her body across perpendicular to the river and swims he will also be carried down the river at the same time. In the river frame she is swimming directly across the river. In the earth frame she is going across the river and being carries down stream. The two velocities have to be added to find her velocity in the earth frame. v " = v " + v " If she wanted to go directly across the river in the Earth frame she would have to point her body at and angle partially upriver. 5) Newton s 1 st law of motion Newton s laws of motion help us to understand, dynamics, the causes of motion of an object. Newton s 1 st law; An object will maintain a constant motion (with constant velocity) until it is acted upon by a net force. An object has inertia, a property that resists changes in motion. We will examine Newton s laws in Inertial reference frames frames that are moving at constant velocity. In a non inertial frame reference frame, an accelerating reference frame like the inside of a rocket ship, objects feel forces. A person in a rocket ship is accelerating and is pressed back against their seat or if there is no seat moves backward in the ship. A person in a car going around a sharp curve is accelerating and is thrown sideways toward the door. We consider inertial reference frames to study Newton s laws because other reference frames have additional forces associated with them. The surface of the Earth is not really an inertial reference system. However, for practical purposes it often serves as one. The Earth is rotating, orbiting the sun, orbiting with the sun around the galaxy and the universe is expanding including a acceleration in the expansion. However, the centripetal accelerations and linear acceleration above are so small that they are not typically noticeable. Also the force of gravity is present but, if you are standing on the surface, does not cause a motion because the surface supports you. However, for very précise experiments you would have to start considering these factors. 6) Forces

4 Force is a vector quantity. If there are several forces acting on an object you add them as vectors to find the net effect. F "# = F Types of forces Contact: Pulling or pushing an object Examples: Direct contact: pulling, pushing or kicking an object Tension: force in rope or other object that transmits a contact force such as pulling to another object. Normal: Floor pushing you upward, opposing the force of gravity, which would otherwise make you fall downward. Friction: Force from a rough surface that resists pushing an object across it. Field: Gravitational force Electric force Magnetic force (Electricity and magnetism are actually just one force known as the Electromagnetic force) There are also two other field forces Strong force (holds the protons and neutrons together in atoms) Weak force (causes weak nuclear decay) Note that all in all contact forces, the thing making the actual contact is electric and magnetic fields of the atoms involved so at the microscopic level all forces are caused by fields. Further the fields are actually caused by either exchanging particles, the messengers of the forces, back an forth (electricity, magnetism, and possibly gravity) or distorting the space around massive objects (gravity). What is the actual cause of gravity is still being debated. However, for practical purposes considering things at the microscopic scale is not necessary. 7) Newton s second law At any moment, the acceleration of an object is proportional to the total (net) force and inversely proportional to its own mass. a = F m

5 F = ma Here: Force and accelerations are vectors. The direction of the net force tells you the direction of the acceleration. Mass is a scalar: If you were pushing several objects at the same time you would add their masses to determine a total mass to use in calculating the acceleration. As vectors we can break the forces into components or reorient the coordinate system to reduce a problem to 2D or 1D in order to make it easier to solve. 8) Gravity We have already extensively considered the gravitational force and the accelerations it causes. F = mg where F is the magnitude of the gravitational force. a = F m g = a = 9.8 m/s, a points downward Note the difference between mass, the quantity used along with g to determine the magnitude gravitational force, and weight, the magnitude of the gravitational force. On the moon there is a different, smaller, gravitational constant, g ""#, and thus a smaller weight, but the mass of an object is the same in both places. Mass is the fundamental property of the object.

Experiencing Acceleration: The backward force you feel when your car accelerates is caused by your body's inertia. Chapter 3.3

Experiencing Acceleration: The backward force you feel when your car accelerates is caused by your body's inertia. Chapter 3.3 Feeling of apparent weight: Caused your body's reaction to the push that the