Projectiles, Circular & Harmonic Motions

• The motion of objects, which move through curved paths, is the result of the body having **components** to its velocity, which are two dimensional in nature. The two dimensions of motion are *independently analyzed* and combined to describe the actual path of the body.

• Independence of the motions is the key to the understanding and simple mathematical analysis of curved motion.

• Symmetry of events is often a useful tool in the analysis of this sort of motion. Here the symmetry is that the path of a body moving through a regular two-dimensional path is the same at the beginning and end of the event.

• Galileo and others, in observing the motions of falling objects from moving platforms, concluded that projectiles and objects dropped from moving platforms have exactly the same characteristics of their paths.

• An object, which is launched and proceeds through space ** freely falling only under the influence of gravity**, is called a

• Projectile motion is classified and dealt with in this unit as either launched horizontally or launched upward at an angle.

• The path through which a projectile moves is called its **trajectory**. The shape of the trajectory of a projectile (neglecting air friction) is a **parabola**. A projectile displays symmetry in its trajectory if the effects of friction are ignored.

• As the name implies, the motion of a body moving in a circular path is called circular motion. A body does not need to move through a complete circle in order to undergo circular motion. Motion through an arc **with a fixed radius** also constitutes circular motion.

• A body can move through a horizontal circle at **constant speed**. This is called *uniform circular motion* and the body undergoes **constant acceleration**, even though the speed remains constant. The direction of the velocity vector constantly changes and thus the body is accelerating.

• The acceleration of a body undergoing circular motion is called the **centripetal acceleration** (**a**_{c}). Equations: **a**_{c} = v^{2}/r (where v is the instantaneous, but constant speed, preferably in m/s, and r is the radius of the circle) and **a**_{c} = 4π^{2}r/T^{2} (where T is the period of one revolution).

• The force, which causes centripetal acceleration, is called the **centripetal force** (**F**_{c}). The centripetal force is the force, acting toward the center of the circle, responsible for causing the centripetal acceleration. Equations: **F**_{c} = *m***a**_{c} (where *m* is the mass of the body) or **F**_{c} = *m* v^{2}/r and **F**_{c} = m4π^{2}r/T^{2}. Fighter pilots making high-speed turns experience large centripetal forces which can sometimes cause them to lose consciousness. The centripetal force divided by *g* (9.8 m/s^{2}) is referred to as *g forces*. Humans in good condition can tolerate up to 9 *g's* without physical harm (except for a brief loss of consciousness).

**Simple harmonic motion** is linear motion in which the **acceleration is proportional to the distance from the equilibrium position and is directed toward that position.** Examples of harmonic motion are a swinging pendulum and a weight bobbing up and down on a spring.

• **Periodic motion** occurs when a body repeatedly moves over the same path in equal intervals of time.

• A **pendulum** is an object suspended so that it can swing back and forth about an axis. The motion of a pendulum is independent of its mass. The motion of a pendulum is independent of its amplitude (in swings of 100 or less?). The period of a pendulum is directly proportional to the square root of its length. The period of a pendulum is inversely proportional to the square root of the local acceleration of gravity (and so is sometimes used to measure local gravity). The pendulum equation:

• **Frequency** is the **number of cycles that an object in periodic motion will complete in a set period of time (preferably seconds).** The unit for frequency (ν) is **Hertz** (Hz), which stands for cycles per second. The reciprocal of frequency is the **period** (T): T = 1/ν. The period is the number of seconds it takes to complete one complete cycle of periodic motion.

- Divers at Acapulco dive from a cliff that is 61 m high. If the rocks below the cliff extend outward for 23 m, what is the minimum horizontal velocity a diver must have to clear the rocks safely?
- A dart player throws a dart horizontally at a speed of 12.4 m/s. The dart hits the board 0.32 m below the height from which it was thrown. How far away is the player from the board?
- A plane flying horizontally at 350 m/s releases a package at an altitude of 1500-m.
- How long will the package take to reach the ground?
- How far will it move horizontally while falling?

- In a TV tube, electrons are projected at a speed of 9.5 x 10
^{5}m/s toward the screen 0.45-m. away. If gravity were the only force acting, how far would the electron fall when it hits the screen?

- An arrow is shot at a 30.0° angle with the horizontal. It has a velocity of 49 m/s.
- How high will the arrow go?
- What horizontal distance will it travel?

- In setting up a chase scene for a movie, a stunt coordinator builds a 45° take off ramp. If a car is to be driven off the ramp at a speed of 50.0 m/s, how far away should the landing ramp be placed?
- A pitched ball is hit by a batter at a 45° angle. It just clears the outfield fence, 98-m away. Find the velocity of the ball when it left the bat. Assume the fence is the same height as the pitch.
- A basketball player tries to make a half court jump shot, releasing the ball at the height of the basket. Assuming the ball is launched at 51.0°, 14.0 m from the basket, what velocity must the player give the ball?
- A baseball is hit at 30.0 m/s at an angle of 53° with the horizontal. Immediately an outfielder runs 4.00 m/s toward the infield and catches the ball at the same height it was hit. What was the original distance between the batter and the outfielder?

- An athlete whirls a 7.00kg hammer tied to the end of a 1.3m chain in a horizontal circle. The hammer moves at the rate of 1.0 RPS.
- What is the centripetal acceleration of the hammer?
- What is the tension in the chain?

- Sue whirls a yo-yo in a horizontal circle. The yo-yo has a mass of 0.20 kg and is attached to a string 0.80 m long.
- If the yo-yo makes 1.0 complete revolution each second, what force does the string exert on it?
- If Sue increases the speed of the yo-yo to 2.0 revolutions per second, what force does the string now exert?

- A 5.0-g coin is placed on a stereo record revolving at 33 1/3 RPM.
- In what direction is the acceleration of the coin, if any?
- Find the acceleration of the coin when it is placed 5.0, 10, and 15 cm from the center of the record. What force accelerates the coin?

- According to the
, (1990 edition, p. 169) the highest rotary speed ever attained was 2010 m/s (4500 mph). The rotating rod was 15.3 cm (6 in) long. Assume the speed quoted is that of the end of the rod.*Guinness Book of World Records*- What is the centripetal acceleration of the end of the rod?
- If you were to attach a 1.00g object to the end of the rod, what force would be needed to hold it on the rod?
- What is the period of rotation of the rod?

- A ride called the
*rotor*has a 2.0m radius and rotates 1.1 RPS.- Find the speed of a rider on the rim of the rotor.
- Find the centripetal acceleration of the rider.

- It takes a 615kg racing car 14.3 s to travel at a uniform speed around a circular racetrack of 50.0-m radius.
- What is the acceleration of the car?
- What average force must the track exert on the tires to produce this acceleration?

- A pendulum makes 35 complete oscillations in 12 s.
- What is its period?
- What is its frequency?

- A pendulum is 3.500 m long. What is its period at the North Pole? (Where the acceleration of gravity is 9.832 m/s
^{2}). What is its period in Java? (where the acceleration of gravity is 9.782 m/s^{2}) - A pendulum has a frequency of 5.50 Hz on earth at a point where g = 9.80 m/s
^{2}. What would be its frequency on Jupiter? (where the acceleration of gravity is 2.54 times that on earth) - A pendulum extends from the roof of a building almost to the floor. If the pendulum's period is 8.5 s, how tall is the building?

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