Circular motion (DONE) Flashcards
What is 1 radian and what analogy can be used to help describe it?
- Radian is the SI unit for angles.
- The angle at which the arc length = the radius is 1 radian.
- If you have a cake and cut a slice, you can measure around the edge of the slice (the arc length) and see how it relates to the radius.
- You will find that if the arc length and the radius are the same amount, then the angle theta = 1 radian.
How can you calculate the number of radians in a full circle?
- We know that the circumference c of a circle is equal to 2pier
- We also know that the radian is equal to the ratio of the arc length to the radius.
- So we know that the number of radians = (2pier)/r
- In this full circle the two r values cancel leaving radians = 2pie rad in one circle.
- This means there is 1 pie rad in a half circle.
How do you work out how many degrees are in a radian?
- To work out how many degrees a radian would be, you would say 1 pie rad = 180 degrees.
- Then divide both sides by pie to find 1 rad = 57.3 degrees.
How will the angular velocity and regular velocity of objects on a rotating turntable vary?
- Anything on the turntable will have identical angular velocities, they move through the same angle each second.
- However if you had an object close to the centre and another object close to the edge, using speed = distance/time you can work out that the object closest to the edge will have a higher velocity.
What is angular velocity?
- Angular velocity is the angle something moves through each second, it has the symbol omega, w.
What units does angular velocity have?
- With pie being a number and T with units s^-1, angular velocity w should have units s^-1.
- However we need to be specific that it is based on the radians it moves through each second so the units for angular velocity is rad s^-1,
- Rad is a dimensionless quantity it just show that we are not talking about degrees.
What are the equations for angular velocity?
- w = 2pie/T
- w = 2pie f
- For number 2 the units for frequency is hertz which is again s^-1.
What is the relationship between velocity and radius from the rotational centre?
- The velocity of objects rotating further from the centre is higher meaning that velocity is directly proportional to the radius.
What is the relationship between velocity and angular velocity?
- The velocity is directly proportional to the angular velocity.
How can we derive an equation for the velocity of objects on a rotating surface?
- velocity is directly proportional to both radius and angular velocity.
- therefore v = wr
How could you calculate the angular velocity of a clock hand?
- Using the example of a clock, if you look at the seconds hand you can calculate the angular velocity.
- The angular velocity w = 2pie/T
- Where T is the time for one complete rotation in seconds.
- Therefore w = 2pie/60 = 0.105 rad s^-1.
- When looking at the minute hand w = 2pie/60x60 = 1.75 x 10^-3 rad s^-1
How could you calculate the angular velocity of a car engine for 3000rpm?
- When using the example of a car, 3000rpm is the number of revolutions per minute.
- Therefore w = 3000 x 2pie/60 = 314 rad s^-1
What types of force may cause an object to experience a centripetal force and acceleration?
- Using newtons first law, if there is a force acting in the direction of motion then the magnitude of velocity will be impacted.
- If a force acts 90 degrees to the motion the force will not impact the velocity however it will cause an acceleration as it changes the direction of motion.
- Provided the force remains at 90 degrees to the motion of the object, the force will result in a centripetal acceleration causing the object to move in a circular parabolic path.
What is the equation for the centripetal acceleration?
- The centripetal acceleration, a = v^2/r.
- However this equation can be varied, the velocity of an object, v = wr.
- Therefore we can substitute this into the centripetal acceleration a, equation.
- A = w^2r^2/r
- Meaning the final equations for centripetal acceleration can be written as:
a = v^2/r
or
a = w^2r
How can we derive an equation for the centripetal force?
- For an object with a constant mass, we know the force on the object, F = ma.
- Using the centripetal acceleration a = v^2/r, we can substitute this into F = ma.
- This means for an object undergoing circular motion the force, F = (mv^2)/r
- We can also use another equation for the centripetal force.
- This is because the centripetal acceleration is also, a = w^2*r.
- We can substitute a into the equation to get centripetal force, F = mw^2*r.
What are real life example of objects experiencing a centripetal force?
- If we think about the sun, the centripetal force is provided by gravitational attraction.
- Electrons could experience a centripetal force provided by electrostatic forces of attraction from the nucleus.
- A car turning a corner experiences a centripetal force provided by friction between the tires and the road.
What are real life examples of objects moving in circular motion with the centripetal force acting at an angle?
- In for example a NASCAR race the sides will be banked and at an angle, so when the vehicles are moving around they are moving at an angle.
- This is the same for an airplane turning around in the air.
- Imagine that a plane stays at the same altitude whilst it turns at an angle in a circular motion.
What forces are acting on a plane at an angle which undergoing circular motion in air whilst maintaining the same altitude?
- A free body diagram will show there are only 2 forces acting on the plane, with the weight acting downwards and the lift force acting perpendicular to the angle at which the plane is banked at.
- The lift force has a vertical and horizontal component.
- The horizontal component acts towards centre.
- In order for the plane to maintain altitude, the vertical component of the lift force must be equal to the weight.
- Through using trigonometry the angle at which the plane is banked can be calculated.
How can you work out the angle an object such as a plane needs to be banked at to undergo a certain circular path?
- For a plane you firstly need to use trig to derive equations for vertical and horizontal components.
- Vertical component = Lcos x theta and the horizontal component = Lsin x theta where L is the lift force and theta is the angle between the direction of the lift force and the vertical force.
- The weight acting down, F = mg.
- The vertical component of lift force needs to = weight in order to maintain altitude.
- Therefore mg = Lcos x theta
- This can be rearranged to L = (mg)/costheta
- The horizontal force acting inwards must provide the centripetal force to allow the plane to take a circular path.
- We know centripetal force F = (mv^2)/r and also that F = Lsin x theta
- Therefore Lsin x theta = (mv^2)/r
- You can then replace the āLā term with (mg)/costheta.
- This will then be [(mg)/costheta]sin x theta = (mv^2)/r
- Then through cancelling and using trig rule sin theta/cos theta = tan theta, we get tan x theta x g = v^2/r
- Therefore tan x theta = v^2/(gr).
- tan theta is the angle at which the plane is banked at.