Topic 6 Circular Motion and Gravitation LEARNING OBJECTIVES Topic 6 The Killers 1. Centripetal Force 2. Newton s Law of Gravitation 3. Gravitational Field Strength
ROOKIE MISTAKE! Always remember. the formulae for centripetal force and gravitational fore are equal when considering orbits. 1 Centripetal Force Radians Measuring angles in degrees becomes less useful in advanced maths and physics, because they are arbitrary. Radians are used because they are multiples of π, which is a natural number and the natural unit for trigonometric functions. The angle in radians, θ, is defined as the arc-length divided by the radius.
Radians For a complete circle: θ = arc-length = 2πr = 2π rads radius r Any other angles are a fraction of 2π Degrees and Radians Degrees 90 180 π/2 Radians π 360 45 2π π/4 Angular Displacement, θ Angular displacement, θ, is the angle in radians (or degrees) through which a point has been rotated about a specified axis. Angular Displacement, θ
Linear Velocity Linear velocity (or tangential velocity) is the velocity of a point on a rotating object. It is given by the equation: v = 2πr t Where r is the radius of the point and t is the time taken for one revolution. Units of v are ms -1 Linear Velocity Consider a rotating bicycle wheel: The radius is smaller for the red point, but the time taken for one complete revolution remains the same. vred < vblue Angular Velocity Angular velocity is the rate of change of angular displacement with respect to time It is given by the equation: ω = θ t Where θ is the angular displacement and t is the time taken. Units of ω are rads -1
Angular Velocity Consider the rotating bicycle wheel again: As the wheel rotates, the angle subtended by both red and blue points is the same, with respect to time. ωred = ωblue Linear and Angular Velocity The relationship linking linear (tangential) and angular velocity is: v=ωr Velocity Changes When an object travels at a constant speed in circular motion, the velocity is constantly changing due to the constant change in direction. If velocity is changing then the object is accelerating
Centripetal Acceleration Acceleration is defined as the rate of change of velocity. Consider the change of direction of velocity below: When an object is in circular motion, the acceleration of the object is always directed towards the centre of the circle. ACTUAL EXAMINER FEEDBACK Candidates were unable to use a vector diagram to explain the need for a centripetal force in circular motion. Acceleration Expressions Centripetal Acceleration a = v r 2 Angular Acceleration a = ω 2 r measured in ms -2 measured in rads -2
Centripetal Force Newton s 2nd Law states: Resultant force equals mass times acceleration in the direction if the force. An object in circular motion always experiences a force directed towards the centre of the circle. Force Expressions Using Centripetal Acceleration F = ma = mv 2 r Using Angular Acceleration F = ma = mω 2 r Measured in Newtons, N Topic 6 The Killers 1. Centripetal Force 2. Newton s Law of Gravitation
Gravitational Force Gravitational force acts at a distance and has an associated force field Newtons 3rd Law: Earth exerts a gravitational force on the Moon. Moon exerts an equal and opposite gravitational force on the Earth. Gravity is the weakest force and is only measurable with large masses (e.g. planet sizes) Gravitational Force The size of the gravitational force is: Newton s Law of Gravitation, where: ACTUAL EXAMINER FEEDBACK Candidates could not provide a statement to encompass Newton s Law of Gravitation
Gravitational Force The size of the gravitational force is: Directly proportional to the product of the masses Indirectly proportional to the square of the distance between the masses Assumption: masses have uniform density and the mass is concentrated at the centre (i.e. point masses) Orbits The size of the gravitational force is EQUAL TO the centripetal force Kepler s Law Not directly examined BUT Recent paper has asked to prove: WHAT?!?
Kepler s Law Not directly examined BUT Recent paper has asked to prove: Start Here: Substitute into above: rearrange and. Topic 6 The Killers 1. Centripetal Force 2. Newton s Law of Gravitation 3. Gravitational Field Strength Gravitational Field Strength A test mass is needed to define the strength of the gravitational field surrounding a large mass The test mass must have negligible gravitational effect i.e. very small mass Gravitational field strength is the force per unit mass experienced by a small test mass:
Gravitational Field Strength Gravitational field strength is the force per unit mass experienced by a small test mass: Gravitational Field Lines Gravitational field strength is a vector Gravity is always attractive Find lines point towards the centre of the massive object Field lines are perpendicular to the surface Lines are closer together near the surface - g is greater near the surface. Line are further apart as distance increases from the surface - g decreases with distance. Special Relationship Outside the spherical mass: g is inversely proportional to the r 2 (distance from centre of mass) Inside the spherical mass: g is directly proportional to r from the centre of the mass (assuming uniform density).
Two Bodies The syllabus says the students should be able to determine the resultant gravitational field strength due to two bodies along a straight line. Question 3m 3m 6m Calculate the resultant gravitational field at P.