MOTION IN TWO DIMENSIONS Some diagrams and simulations from

Newton's laws help to explain the motion (and specifically, the changes in the state of motion) of objects which are either at rest or moving in 1-dimension. PROJECTILE: is an object upon which the only force acting is gravity. One of the most common example of an object which is moving in two-dimensions is a projectile. Projectile Motion

EXAMPLES OF PROJECTILES an object which is thrown upwards at an angle is also a projectile an object dropped from rest is a projectile an object which is thrown vertically upwards is also a projectile

MISCONCEPTION: Some people think that a projectile must have a force acting upward upon it in order for it to climb. This is not true. A force is not required to keep an object in motion. A force is only required to maintain acceleration. The path of a projectile is not that complicated. Many projectiles not only undergo a vertical motion, but also undergo a horizontal motion. The horizontal and vertical components of the projectile’s motion are independent of each other.

Vector Combination

Horizontal MotionVertical Motion Forces NoYes The force of gravity acts downward Acceleration NoYes "g" is downward at ~ -9.8 m/s/s Velocity ConstantChanging (by ~ 9.8 m/s each second)

WHAT IF THE PROJECTILE IS SHOT UPWARD?

Now assume gravity is turned on The projectile would travel with a parabolic trajectory. The downward force of gravity will act upon the cannonball to cause the same vertical motion as before - a downward acceleration. The cannonball falls the same amount of distance in every second as it did when it was merely dropped from rest.

Projection Angles With the same initial speed but different projection angles, a projectile will reach different altitudes (height above the ground) and different ranges (distances traveled horizontally). However, the same range can be obtained from two different angles, symmetrically around a maximum of 45˚, as shown in the graph.

Symmetry The path of a projectile is symmetrical… It rises to its maximum height in the same time it takes to fall from that height to the ground. Because acceleration is the same all of the time, the speed it loses while going up is the same as the speed it gains while falling. Therefore the speeds are the same at equal distances from the maximum height, where the vertical speed is zero.

Question: They strike at the same time. At the instant a horizontally held rifle is fired over level ground, a bullet held at the side of the rifle is released and drops to the ground. Ignoring air resistance, which bullet strikes the ground first?

Question: Vertical -9.8 m/s 2 Horizontal 0 m/s 2 A projectile is launched at an angle into the air. If air resistance is negligible, what is the acceleration of its vertical component of motion? Of its horizontal component of motion?

Question: At what part of its trajectory does a projectile have minimum speed? At maximum height.

Question: A ball tossed into the air undergoes acceleration while it follows a parabolic path. When the sun is directly overhead, does the shadow of the ball across the field also accelerate? NO

Question: How can an object be moving upward if the only force acting upon it is gravity? Newton’s first law

Question: What launch angle maximizes the range (horizontal distance)? 45°

Question: What launch angle maximizes the height reached? 90°

Question: How does the time spent in the air depend on the launch angle? Time is maximized at 90°

Question: Compare what happens at complementary launch angles. Same range (horizontal displacement)

Question: What happens to the trajectory when the mass of the projectile is changed? Nothing

Question: Compare the trajectories of a projectile that is under the influence of gravity and one that is not? NO Gravity (straight line) Gravity (parabolic trajectory)

Simulation #1

Simulation #2

Q’s 1-4 of Handout Study Guide 7.1 Q’s 4-8 of Handout Q’s 1-8 pg 536 & 537 (pdf 67) Do

Periodic Motion

Projectile motion is two-dimensional, but it does not repeat. Projectiles do not move all along their trajectories more than once. Periodic motion can also be an example of 2 dimensional motion however it involves motion that repeats itself at regular intervals. Examples of periodic motion are a yo-yo being swung horizontally overhead, an object bouncing on a spring or the pendulum of a clock.

Circular motion An object that moves in a circle at constant speed is said to experience UNIFORM CIRCULAR MOTION. Recall that velocity is a vector quantity so it has both magnitude and direction. With circular motion an object may have a constant speed but a direction that is constantly changing. If this is the case then the object is said to be undergoing centripetal acceleration.

Centripetal acceleration: centripetal means center seeking, or acceleration in the direction of the center of the circle and can be found using the formula Units? Centripetal acceleration always points towards the center of the circle and is directly proportional to the square of the speed and inversely proportional to the radius of the circle.

The velocity of an object undergoing circular motion can be found using the following formula, Where the distance traveled is equal to the circumference of the circle, and the total time is one period. PERIOD (T) - the length of time needed to complete one cycle of motion.

by combining the two above equations we get a second formula for a c

Newton’s second law tells us that an object does not accelerate unless there is a force that acts on it. For circular motion this force is called a centripetal force and is also directed radially inwards.

This diagram shows the centripetal force acting in the same direction as the acceleration. If you remove the centripetal force, the object will not continue moving in a circular path. Examples: merry-go-round in the park, tilt-a-whirl, Nascar banking on the turn at Talledega

Example: A kg rubber stopper is attached to a 0.93 m length of string. The stopper is swung in a horizontal circle, making one revolution in 1.18 seconds. a) Find the speed of the stopper. b) Find its centripetal acceleration. c) Find the force that the string exerts on it. a) 5.0 m/s, b) 26 m/s 2, c) 0.34 N

Ring around the collar Q’s 9-15 Handout Q’s pg 559 (pdf 68) omit # 17 Do

Simple Harmonic Motion Simple harmonic motion is another example of periodic motion. The key thing with simple harmonic motion is that there must be a restoring force that causes the object to return to the equilibrium position.

The restoring force must vary linearly with respect to displacement. ie: small displacement, small force large displacement, large force The maximum displacement is called amplitude. Amplitude: the maximum distance that the object moves from its equilibrium position.

An example of a type of simple harmonic motion would be that of a pendulum. Note that the restoring force here is the tangential component of the weight and that it increases with amplitude.

Similar to the pendulum, the spring is yet another example of SHM where the restoring force varies linearly with respect to displacement and is always in the direction of the equilibrium position.

The period of a simple pendulum and that of a spring can be found by the following formulas, PendulumSpring

Example: On top of a mountain a pendulum 1.55 m long has a period of 2.51 s. What is the acceleration due to gravity at this location? 9.71 m/s 2

Example: A 500 g mass on a spring is displaced by 4.75 cm from its rest position and is allowed to oscillate. If the period is measured to be 2.3 sec calculate the spring constant of the spring. 3.7

Example: Find the length of a pendulum oscillating on the moon that would have the same period as a 1.0 m long pendulum on earth. Assuming that the gravity on the moon is 1 sixth that of earth’s m

Example: A square pig is seen bouncing on a pogo stick with a frequency on 0.54 Hz. If the pogo stick company advertises that their sticks have a spring constant of 290, what is the mass of the pig? 25 kg

Q’s 1&2 pg 608 (pdf 75) Q’s 5-8 pg 614 (pdf 75) Study Giude7.2 Do

Torque and Static Equilibrium

From the chapter on forces we know that if the net force on an object is zero than the objects net acceleration must be zero. if F net = 0 than a = 0

Until now we have always assumed that all forces acted on the center of the object. Now we will look at what would happen if the forces were located at some distance from the center.

In fact the object would begin to rotate around it’s center of mass. At first glance this may seem to violate Newton’s laws of motion because the object begins to rotate even though the net force is zero. However if you take notice of the center of mass of the object, it does not move. This is one of the key differences between translational motion (center of mass moves) and rotational motion (object rotate around the center of mass)

The force that induces rotational motion is called torque. A positive torque will cause a counterclockwise rotation and a negative torque will cause a clockwise rotation. Torque is measured in Newton meters (Nm), has the symbol tow (τ) and can be found using the following formula. where r is the distance from the center of mass, or pivot point, to the position where the tangential component of the force is being applied.

Example: Cuddals, a soft and cuddly Rottweiler, wants to go outside. He pushes on the door with a 45 N force at and angle of 5° from the perpendicular, 60.0 cm from the hinges. What perpendicular force is he applying to the door and what is the final torque? 44.8 N, 26.9 Nm

Example: Two girls are applying torque to steering wheel (40.0 cm in diameter) of the bumper car during an amusement park ride. The girl on the left applies a force of 10.0 N up, while the girl to write pulls directly down with a force of 15 N. What is the net torque on the steering wheel? 5 Nm

Conditions for Static Equilibrium For an object to be truly motionless, it must have a net force of zero but it also must have a net torque of zero. F net = 0 & Γ net = 0

Example: A cable is attached to the shaft of a 395 kg crane at a point 5.0 m from the hinged (pivot) point. When the crane is horizontal, the cable makes a 55 degree angle the crane. if the center of mass is located halfway between the pivot point and the cable, and the Crane is in static equilibrium what is the tension in the cable? 2400 N