Physics Mod 5 Summary PDF

**[Projectile Motion:]** How can models that are used to explain projectile motion be used to analyse and make predictions? - Projectiles launched horizontally - Projectiles launched obliquely **What is a projectile?** - A projectile is any object that is thrown or projected into the air...

**[Projectile Motion:]** How can models that are used to explain projectile motion be used to analyse and make predictions? - Projectiles launched horizontally - Projectiles launched obliquely **What is a projectile?** - A projectile is any object that is thrown or projected into the air and is moving freely -- this means that it doesn't have a power source (engine/propellor) **What type of flight path does a projectile follow?** - Projectiles always follow a ballistic flight path -- a parabolic path **Which forces act on a projectile while in motion?** - While there may be a forward force required to launch the projectile, once released this force no longer acts upon it. The only force acting on the projectile is the weight force (the force due to gravity = F~g~ = *mg*) **What can you use to solve problems with projectiles launched obliquely?** - When launched at an angle to the horizontal, use trigonometry to determine the initial horizontal and vertical velocities - When an oblique projectile is at its highest point, the projectile is only moving horizontally, and thus its velocity is given by the horizontal component of its launch velocity (the vertical component of the velocity is zero at this point) **How do you solve projectile questions?** 1. Draw a diagram 2. Separate vertical and horizontal components 3. Horizontal movement: Neglecting air resistance leads to no horizontal acceleration, causing initial and final velocity to be equal (a~x~ = 0) (*u~x~=v~x~*) 4. Vertical movement: No initial velocity, thus acceleration (when on earth!) is due to gravity, thus always 9.8ms^-1^ downwards (-9.8ms^-1^), at the point of maximum height v~y~ = 0 (as only moving horizontally) 5. Time: the same value of time applies to both horizontal and vertical, this value is only impacted by the height and angle of the projectile -- not the velocity **[Circular Motion:]** Why do objects move in circles? - Circular motion - Circular motion on banked tracks - Work and Energy - Torque **What is uniform circular motion?** - Uniform circular motion involves an object travelling in a circular path at a constant speed and continually changing velocity (as direction changes), the velocity of the object is always tangential to its path -- due to changing velocity, the object is accelerating **What is the relationship between period and frequency?** - T = 1/f OR f = 1/T (f is the frequency in hertz, T is the period in seconds) - Period is the amount of time taken to travel around the circle once (one complete revolution), frequency is the number of rotations completed each second **What is the instantaneous velocity (AKA average speed) of an object moving in circular motion?** - Average speed is equal to the speed / time = circumference / period - Thus, to determine the average speed, divide the distance travelled in one revolution (circumference = 2πr) by the time taken (period = T) which gives the formula 2πr/T -- radius (m), v = speed (ms^-1^), T = period (s) **How to measure the angular velocity and why is this used?** - Angular velocity is used to determine the angle of rotation in a given time: - [\$\\omega = \\ \\frac{\\Delta\\theta}{t}\$]{.math.inline} where t = time, theta = angle in radians and units given in radians/second (the change in theta is equal to the angle travelled) **What formula is the result of merging the formulas for both angular and instantaneous velocities?** - [*v* = *ωr*]{.math.inline} -- THIS FORMULA ISN'T ON THE FORMULA SHEET! **What are polar coordinates?** - (r, θ) where r = distance from centre (radius) and θ = angle in radians **What is centripetal acceleration and its formula?** - Centripetal acceleration always occurs towards the centre of the circle (think of it as gravity pulling it in, however it's not always gravity) - [\$a\_{c} = \\frac{v\^{2}}{r}\$]{.math.inline} **Describe and define centripetal force and its formula:** - The centripetal force isn't an individual force like gravity, but rather whichever force in a system keeps the object going in a circle. It is defined as the amount of force needed to keep it going in a circle (there can be more, but this is what is required to balance the normal force and gravity) - Fc = m x a~c~ - [\$F\_{c} = \\frac{\\text{mv}}{r}\$]{.math.inline} **Explain the forces acting upon a conical pendulum:** - In a conical pendulum, a mass travels in a horizontal circle on the string -- the tension in the string supplies the force to counter gravity and centripetal force. The forces acting upon the masss are Tension, mg and Fc. The centripetal force is whichever force is keeping the ball moving in a circle. Note that a smaller radius makes a smaller angle between the centre and circumference. The following formulas are used to solve these problems: - [\$a\_{c} = \\frac{v\^{2}}{r}\$]{.math.inline} - [\$v = \\frac{2\\pi r}{T}\$]{.math.inline} - [\$F\_{c} = \\frac{mv\^{2}}{r}\$]{.math.inline} - [\$\\omega = \\frac{\\mathrm{\\Delta}\\theta}{t}\$]{.math.inline} **How do you find the angle of the string?** - A slower ball swings at an angle closer to the pole whilst a faster ball swings closer to the horizontal (can't reach perfectly perpendicular). To find this angle, use the formula: \ [\$\$tan\\theta = \\frac{\\text{opp}}{\\text{adj}}\$\$]{.math.display}\ **Explain the Concept of Banked Tracks:** - Banked tracks are tracks that are inclined at some angle to the horizontal (use same diagram as conical pendulums, but normal force instead of tension force) **Describe what occurs as a car drives on a banked track:** - As the car travels in a straight line, it hits the wall in front of it which causes the acceleration to act towards the centre of the circle (banked tracks don't always have to be full circles, not have circular ends) **Why are banked tracks used?** - Banked tracks allow for greater speed. An increase in speed increases the centripetal force, which the normal force (due to the bank) helps to counterbalance. They also eliminate the need for sideways friction when turning **How do you find the angle of bank (at design speed)?** - [\$tan\\theta = \\frac{F\_{\\text{net}}}{F\_{g}}\$]{.math.inline} F~net~ is the force acting towards the centre of the circle, F~g~ is gravity - [\$tan\\theta = \\frac{v\^{2}}{\\text{rg}}\$]{.math.inline} v = vehicle speed, g = magnitude of acceleration due to gravity (9.8), r = track radius (m) - [\$v = \\sqrt{\\text{rg\\ tanθ}}\$]{.math.inline} v = vehicle speed (m/s) = design speed **Torque:** **Define torque:** - Torque is the turning movement of a force (*T*), it is the product of distance and a perpendicular force Remember: Torque isn't a force, rather it's the twisting effect of a force [*τ* = *r*⊥*F* = *r* *F* *sinθ*]{.math.inline} -- θ is the angle between the radius vector and force vector **[Motion in Gravitational Fields:]** How does the force of gravity determine the motion of planets and satellites? - Gravity - Satellite Motion - Gravitational Potential Energy **M is being orbited, m is orbiting** **Explain Newton's Law of Universal Gravitation:** - Every object is attracted to every other object -- always attractive as gravity can't repel (always +ve) - [\$F = \\ \\frac{\\text{GMm}}{r\^{2}}\$]{.math.inline} F = gravitational force (N), M and m = masses of 2 objects (kg), r = distance between centres of 2 objects (m), G = gravitational constant = 6.67 x 10^-11^ (Nm^2^ kg^-2^) **What is gravitational field strength?** - Gravitational field strength is also known as the acceleration due to gravity on different planets (always 9.8 m/s on earth) **Explain the concept of orbits:** - For an object (m) in a circular orbit around another object (M), the centripetal force **[is]** the gravitational force - Elliptical orbits have two focal points (further apart = more squashed/eccentric) - Circular orbit: magnitude of velocity is constant (perpendicular to force), when in circular motion the centripetal force = the gravitational force between the 2 objeccts - [\$v = \\ \\sqrt{\\frac{\\text{GM}}{r}}\$]{.math.inline} M = mass of central body (being orbited), r = orbital radius, v = orbital velocity (speed) **Explain the relationships involved with Kepler's Law of Periods:** - Kepler's Law of Periods compares the periods and radii for different satellites orbiting same the central body -- FORMULA IS TRUE FOR ALL UNITS, there are two types of questions: using the formula to find a value or using the relationship (constant when orbiting same central body) - [\$\\frac{r\^{2}}{T\^{3}} = GM\$]{.math.inline} = CONSTANT r = radius, T = period - [\$\\frac{{r\_{1}}\^{3}}{{T\_{1}}\^{2}} = \\frac{{r\_{2}}\^{3}}{{T\_{2}}\^{2}}\$]{.math.inline} **Define a satellite:** - A satellite is something in orbit -- "an object in a stable orbit around a larger central mass" - Low earth satellites (altitude is less than 1000km) have low periods (no lower than 1.5 hrs as atmospheric drag increases largely when r gets even smaller), low radius, large velocity - Geostationary satellites: satellite remains stationary in the sky -- earth orbiting satellite orbits directly above equator (always as 42200km) -- satellite must revolve west to east (same direction as earth) **Low Earth Orbit** **Geostationary** ---------------------- --------------------- ------------------- **Orbital Period** 1.5 -- 4 hrs 24 hrs (on earth) **Altitude** 400 -- 1000 km 35800 km **Orbital Radius** 6800 -- 7400 km 42200 km **Orbital Velocity** 3070 m/s **Explain the concept of Escape Velocity:** - Escape velocity is the speed (not velocity as no direction) an object needs to just escape a planet's gravitational field - [\$v = \\ \\sqrt{\\frac{2\\text{GM}}{r}}\$]{.math.inline} (greater m = greater v, greater r = smaller v) - [\$E = U + k = \\ - \\frac{\\text{GMm}}{2r}\$]{.math.inline} (can't be used as only when in orbit) **Gravitational Potential Energy:** - [\$U = \\ - \\frac{\\text{GMm}}{r}\$]{.math.inline} (negative as when r gets bigger, U gets bigger, and as r approached infinity, u approached 0)