module 5: advanced mechanics Flashcards
What is a projectile?
A projectile is an object that is moving freely under the influence of gravity.
* For an object to be a projectile, there must be no force other than the gravitational force acting on the object.
* The net force on the projectile is the gravitational force (its weight).
The trajectory of a projectile follows a parabolic path.
Horizontal component of projectile motion
Horizontal Motion is constant velocity motion, the velocity is the same as the horizontal component of the initial velocity.
Vertical component of projectile motion
Vertical motion is uniform accelerated motion, the projectile experiences a constant downward acceleration due to gravity.
Projectile situation 1:
Launched from an elevated position and follows a parabolic path to the ground
- Initial velocity is only in the horizontal direction (Initial velocity = horizontal velocity).
- Velocity direction does not have initial velocity (initial vertical velocity = 0 ms-1).
- Launch angle 𝜃 = 0°
Projectile situation 2:
Launched at an angle, rising upward to a peak before falling back down
- Total vertical displacement = 0
- When y is at maximum height, v = 0 ms-1
- Initial speed = final speed
- Time to rise = time to fall
- Initial angle = final angle
Projectile situation 3:
Launched at an angle from an elevated position
- Total vertical displacement = difference in height between the two levels
- When y is at maximum height, v = 0 ms-1
- Time to rise ≠ time to fall
- Initial speed ≠ final speed
- Initial angle ≠ final angle
Circular motion
For an object to move in a circular path, it must continually change the direction its travelling, hence circular motion requires a net force to be applied on the object.
Uniform circular motion refers to objects that move in a circle at constant speed.
Centipetal force
Centripetal force is the force that causes an object to move in a circular path and is directed towards the centre of the circle.
Tangential Velocity
Tangential velocity is the velocity of an object experiencing uniform circular motion.
* Tangential velocity of an object undergoing circular motion is related to the radius of the circle and the time it takes the object to complete one revolution.
Tangential Velocity equation
v = 2πr/T
Where:
v is the tangential velocity (ms-1)
r is the radius (m)
T is the period of rotation i.e. the time to go around the circle once (s)
Centripetal Accleration equation
ac = v^2 / r
Centripetal Force
Any motion in a curved path represents accelerated motion and requires a force directed towards the centre of curvature of the path.
* This force is called centripetal force which means “centre seeking” force.
Centripetal Force equation
FCentripetal = mv^2 / r
Unbanked curve
On an unbanked curve, the static frictional force from the car tires provides the centripetal force.
* If the frictional force is less than the centripetal force required, then the car cannot maintain its circular path and the car will skid at a tangent to the circle.
Banked curve
When the curve is banked, the centripetal force can be partially or entirely supplied by the horizontal component of the normal force.
* A reason to have banked curves would be to decrease the reliance on the force of friction.
* For every banked curve, there is one ideal speed (regardless of mass).
Ideal speed
When centripetal force is entirely supplied by the horizontal component of the normal force, it is called ‘ideal speed’.
* Speed of car > ideal speed, car will slide up inclined plane.
* Speed of car < ideal speed, car will slide down inclined plane.
Angular acceleration
In UCM, angular velocity (⍵) is constant.
* If ⍵ changes with time, angular speed is increasing or decreasing, therefore angular acceleration is present
* For an object to start rotating around an axis, it requires a force to give the object some angular acceleration.
Torque
Torque is the rotational equivalent of linear force, it is a vector quantity.
* Occurs when a force, F, is applied at some displacement, r, from an axis of rotation
* The direction of the torque is perpendicular to the plane of rotation, and the direction is determined using the right hand
* Have to indicate whether the torque turns the object clockwise or anticlockwise
Properties of ellipses
- An ellipse is defined by two focal points, together called foci.
* The sum of the distances to the foci from any point on the ellipse is always constant. - The amount of flattening of the ellipse is called the eccentricity.
* If e = 0, it’s a circle.
* If e = 1, it’s a parabola
* All ellipses have eccentricity in the range: 0 ≤ e ≤ 1. - The longest axis of the ellipse is that major axis while the short is called the minor axis.
* Half the major axis is termed a semi-major axis.
Kepler’s First Law - Law of ellipses/Law of orbits
First law: The planet orbits the sun in elliptical orbits, with the Sun as one of the focal points.
Kepler’s Second Law - Law of equal areas
Second law: the line between a planet and the sun (the radius vector) sweeps out equal areas in equal periods of time.
Kepler’s Third Law - Law of Periods
Third law: The square of a planet’s period, T, is directly proportional to the cube of the semi-major axis of its orbit.
Orbit characteristics
Orbits must be:
* Centred on the centre of the mass of the central object
* Elliptical or circular
Orbit characteristics are:
* Altitude (circular or near circular)
* Nearest and furthest point (e.g. perigee and apogee, if elliptical)
* Inclination (relative to the equator)
Satellite definition
A satellite is an object that orbits a central body.
* A manufactured satellite is called an artificial satellite.
* Planets, asteroids, comets etc. are called natural satellites.
