Circular motion (DONE)

Question 1

What is 1 radian and what analogy can be used to help describe it?

Answer 1

• 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.

Question 2

How can you calculate the number of radians in a full circle?

Answer 2

• 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.

Question 3

How do you work out how many degrees are in a radian?

Answer 3

• 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.

Question 4

How will the angular velocity and regular velocity of objects on a rotating turntable vary?

Answer 4

• 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.

Question 5

What is angular velocity?

Answer 5

• Angular velocity is the angle something moves through each second, it has the symbol omega, w.

Question 6

What units does angular velocity have?

Answer 6

• 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.

Question 7

What are the equations for angular velocity?

Answer 7

1. w = 2pie/T
2. w = 2pie f

• For number 2 the units for frequency is hertz which is again s^-1.

Question 8

What is the relationship between velocity and radius from the rotational centre?

Answer 8

• The velocity of objects rotating further from the centre is higher meaning that velocity is directly proportional to the radius.

Question 9

What is the relationship between velocity and angular velocity?

Answer 9

• The velocity is directly proportional to the angular velocity.

Question 10

How can we derive an equation for the velocity of objects on a rotating surface?

Answer 10

• velocity is directly proportional to both radius and angular velocity.
• therefore v = wr

Question 11

How could you calculate the angular velocity of a clock hand?

Answer 11

• 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

Question 12

How could you calculate the angular velocity of a car engine for 3000rpm?

Answer 12

• 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

Question 13

What types of force may cause an object to experience a centripetal force and acceleration?

Answer 13

• 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.

Question 14

What is the equation for the centripetal acceleration?

Answer 14

• 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

Question 15

How can we derive an equation for the centripetal force?

Answer 15

• 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.

Question 16

What are real life example of objects experiencing a centripetal force?

Answer 16

• 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.

Question 17

What are real life examples of objects moving in circular motion with the centripetal force acting at an angle?

Answer 17

• 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.

Question 18

What forces are acting on a plane at an angle which undergoing circular motion in air whilst maintaining the same altitude?

Answer 18

• 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.

Question 19

How can you work out the angle an object such as a plane needs to be banked at to undergo a certain circular path?

Answer 19

• 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 ‘L’ 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.