Centripetal Acceleration & Projectile Motion 4th 6wks
Centripetal Force and Acceleration Centripetal Acceleration (A C ) is the acceleration of an object towards the center of a curved or circular path. Centripetal comes from the Latin for center seeking For a ball to accelerate towards the center and not follow Newton s 1 st Law, a force must be acting upon it. The force is Centripetal Force, (F C ) which is the force caused by the grip or pull between the object and the surface it is upon causing it to be pulled towards the center of the curve. Centripetal Force is the result of another force such as gravity, tension, friction, the normal force, or the electrostatic force between electrically charged objects.
Centrifugal Force v Centripetal Force Historically, and in everyday speech people talk of the centrifugal force The Centrifugal force is the force that pulls a rotating or spinning object out from the center. Centrifugal comes from the Latin for center fleeing Devices like the washing machine, machines that separate out substances, or those that allow pilots to experience extreme forces in a lab are called centrifuges
Centrifugal Force v Centripetal Force However, there really isn t a centrifugal force. The apparent Centrifugal Force is caused by the object s inertia. If centripetal force were removed, the object would actually travel in a straight line (how it had been moving) due to its inertia Remember, that according to Newton s 1 st Law of Motion the object will take a straight path at a constant velocity unless acted upon by an outside unbalanced force. The Centripetal Force is that force If there were a Centrifugal force pulling away, when centripetal force were removed an object would move straight out and travel in the direction of the centrifugal force. V is the inertia, whereas the curve is the resulting path
Ringling Bros. Circus Globe of Steel Motorcycle Ball
MIT Demo: Centripetal Acceleration and Inertia
Centripetal Acceleration and Inertia
NASA on Centripetal Force
Creating Artificial Gravity
Calculating Centripetal Acceleration Note: Tangential Velocity is how fast a point on a circular object (or an object moving in a circular path) is moving a certain distance from the radius of a circle.
Example Problem #1 An amusement park ride spins at 13 m/s. The ride has a radius of 4 m and the ride takes 2 seconds to make a complete revolution. What is the centripetal acceleration? Given: radius = 4 m tangential velocity = 3 m/s Not needed time = 2 sec a c = a c = a c = (13m s )2 4 m 169 m2 s 2 4 m 1 169 m2 s 2 1 4 m a c = 42.25 m/s 2
Example Problem #2 Jimmy moves around in a merry go round at 2 m/s. If the radius of the merry go round was 4 m, what was the centripetal acceleration? Given: radius = 4 m tangential velocity = 2 m/s a c = (2m s )2 4 m a c = 4 m2 s 2 4 m 1 a c = 4 m2 s 2 1 4 m a c = 1 m/s 2
Example Problem #3 A ball is twirled in a circle with a radius of 2 m. If the centripetal acceleration was 0.5 m/s 2, what was the tangential velocity? Given: radius = 2 m centripetal acceleration = 0.5 m/s 2 V t = a c x r V t = V t = 1 m/s 0.5 m/s 2 x 2 m
Example Problem #3 Centripetal Jimmy has a mass of 80 kg and moves around in a merry go round at 1.5 m/s. If the radius of the merry go round was 4 m, what centripetal force did he experience? Force F c = mv2 r F c = 80 kg (1.5 m s )2 4 m F c = 80 kg x 2.25 m2 s 2 4 m F c = 180 kg x m s 2 4 F c = 45 kg x m/s 2 Remember: 1 N = 1 kg x m/s 2 F c = 45 N
Centripetal Acceleration and the Wall of Death
So far, we have only considered situations that were completely vertical: that is an item that was either pitched straight up in the air or dropped straight down. But what of projectiles that have a horizontal component? Note: for our purposes we will consider the horizontal acceleration to be zero. Once the item has been thrown (and left the hand) or the bullet has been shot (and left the barrel) there is no longer any force pushing it horizontally and it is moving under its own inertia. Air resistance at this conceptual stage will be disregarded, though in an atmosphere it would work against the horizontal inertia if it were considered. Therefore, we will look at ideal situations
Projectile Motion: Football Correlation
A projectile is any object that moves through the air or through space, acted on only by gravity (and air resistance). The motion of a projectile is determined only by the object s initial velocity, launch angle and gravity. 1, 2
Projectile motion is a combination of horizontal motion and vertical motion. The horizontal motion of a projectile is constant because no gravitational force acts horizontally 3, 4
The vertical motion of a projectile is nothing more than free fall with a constant downward acceleration due to gravity. 7
The vertical motion of a projected object is independent of its horizontal motion. 5
Let's say a Wiley coyote runs off a cliff. As he leaves the cliff he has a horizontal velocity. As soon as the coyote leaves the cliff he will experience a vertical force due to gravity.
This force will cause him to start to accelerate in the vertical direction. As he falls he will be going faster and faster in the vertical direction
Y X The horizontal and vertical components of the motion of an object going off a cliff are separate from each other, and can not affect each other. In a lot of books you will see the horizontal component called x and the vertical component called y.
A projectile, once projected, continues in motion by its own inertia and is influenced only by the downward force of gravity. Gravity accelerates objects downwards, but is unable to affect the horizontal motion of a projectile. 8, 9
A projectile moves horizontally with constant velocity while being accelerated vertically. The result is a motion in a curved path. 10
The path of a projectile is called its trajectory. The trajectory of a projectile in free fall is a parabola. The total distance traveled by a projectile is known as its range. 11, 12, 13
Horizontal Shot. If the barrel of a rifle is horizontal to the surface of the earth when fired, the bullet never rises above the barrel, and gravity causes an immediate descent.
An object projected horizontally will reach the ground in the same time as an object dropped vertically. No matter how large the horizontal velocity is, the downward pull of gravity is always the same.
Ball Shoot-n-drop
The cannonball falls the same amount of distance as it did when it was merely dropped from rest
Projectiles at an angle If projectiles are launched at the same speed, but at different angles, the height and range is of the projectile are affected. *Calculating these paths (both their height and range) requires trigonometry and/or intro level calculus, so we won t worry about paths at angles at this time. Though the farthest a projectile will travel horizontally when shot at an angle is at a 45 degree angle.
Vertically launched projectile The horizontal velocity component remains the same size throughout the entire motion of the cannonball.
Sports Trivia Maximum range is achieved if the projectile is fired at an angle of 45 degrees with respect to the horizontal. 18
Isaac Newton s Cannonball In laying out his law of universal gravitation, Newton described a mountain so gigantic that its summit poked into space -- and that's where he placed the giant cannon. No, Newton didn't plan to fire at alien invaders. His orbital cannon was a mere thought experiment explaining how one object might orbit another. Load too little or too much gunpowder into this theoretical super weapon, and the cannonball will either fall back to the Earth's surface or sail off into outer space. Just the right amount of powder, however, and you'd give the cannonball sufficient velocity to fall toward the Earth at the same rate that the planet curves away from it. The cannonball, Newton writes, would continue in free fall all the way around the planet, in effect, orbiting it.
Isaac Newton s Cannonball The earth s curvature is such that the surface falls away below a truly flat horizontal line by about five meters in 8,000 meters (five miles). Recall that five meters is just the vertical distance an initially horizontally moving projectile will fall in the first second of motion. But this implies that if the (horizontal) muzzle velocity were 8,000 meters per second, the downward fall of the cannonball would be just matched by the earth s surface falling away, and it would never hit the ground! This is just the motion, familiar to us now, of a satellite in a low orbit, which travels at about 8,000 meters (five miles) a second, or 18,000 miles per hour. Diagram from Newton s book A Treatise of the System of the World (1687)
Escape Velocity The speed necessary to launch an object up such that it will never come back is well defined for masses because the basics of how gravity works is well understood. This speed is called the escape velocity Note that the escape velocity depends upon both the mass M and radius R of a body and uses Newton's Gravitational Constant G = 6.67 10-11 N m 2 /kg 2. Its value for the Earth is 11,200 m/s (or approximately 25,000 miles per hour). escape velocity: the speed needed to "break free" from the gravitational attraction of a massive body, without further propulsion Escape Velocity depends upon the mass of the body a craft is trying to escape from more mass requires a greater escape velocity One reason for the difficulties in planning missions to Mars is a craft would have to bring sufficient fuel to achieve escape velocity from Mars and get back home. The large amount of fuel and speed required lead to cost the recently scrapped Ares 1 rocket that NASA was going to use to put the Constellation space craft into orbit would have cost an estimated $1.6 billion per flight.
Isaac Newton s Cannonball: ball dropped straight down
Isaac Newton s Cannonball: ball shot horizontally at a speed less than 8,000 m/s (~ 18,000 miles per hour)
Isaac Newton s Cannonball: at orbital velocity of at least 8,000 m/s (or approximately 18,000 miles per hour)
Isaac Newton s Cannonball: ball launched at greater than escape velocity of 11,200 m/s (or approximately 25,000 miles per hour)