Motion in 2 Dimensions Chapter 7

Intro: All curved motion is the result of a force that causes an object to deviate from its straight-line motion. According to Newton’s first law of motion, this is caused by an unbalanced force. From this we can conclude that an object moving in a curved path must be acted upon by a net force. The curved path depends only upon the direction and size of the unbalanced force acting on the object.

7:1 Projectiles Projectile: Any object that is thrown or otherwise projected into the air. The characteristic path followed by a projectile is a parabola and is called its trajectory. Projectile motion describes the movement of a projectile along its trajectory.

What happens to an object after it is shot or thrown? The forward motion is called horizontal motion. Once released, there are no accelerating forces, so horizontal motion remains constant. Gravity begins to act on the projectile, creating vertical motion. Horizontal and vertical motion are completely independent of one another. If you were to throw a ball from shoulder height and another ball is just dropped from the same height, which travels farther horizontally? Which ball hits the ground first?

Horizontal displacement depends on the horizontal velocity and the interval of time that the projectile is in the air. The projectile’s initial vertical velocity and the acceleration of gravity determine time interval, which can be found using: dv = ½ gt2 + vit The horizontal displacement is equal to the product of the horizontal velocity and the time of fall dh = vht dh = horizontal displacement vh = horizontal velocity dv = vertical distance (usually negative)

Example: A stone is thrown horizontally at 15 m/s. It is thrown from the top of a cliff 44 m high. A. How long does it take the stone to reach the bottom of the cliff? B. How far from the base of the cliff does the stone strike the ground?

Assignment: P. 116 #1-6

7:2 Projectile Motion When a projectile is fired at an angle above the horizontal, it will be moving both vertically and horizontally along its trajectory. This velocity is resolved into horizontal and vertical components. When solving problems, the horizontal and vertical motions are treated separately.

Example: A golf ball is hit and leaves the tee with a velocity of 25.0 m/s at 35.0° with respect to the horizontal. What is the horizontal displacement of the ball?

Practice and HW Practice: Pg. 117 # 7-8 HW Pg. 117 #9-11

7:3 Uniform Circular Motion Occurs when a net force, acting on a mass moving at constant speed, changes direction in such a way that the force is always acting at a right angle to the direction in which the mass is moving.

Suppose a hand-controlled object is attached to a string and moves in a horizontal circle. Your hand pulls on the strings and provides an inward radial force that keeps the object moving in its circular path. This force is called centripetal force, Fc. Centripetal = center-seeking.

If the force of the string is removed, the object will fly in a straight line, according to Newton’s first law of motion. The straight line path is tangent to the circle at the point of release.

Centripetal force always acts at a right angle to the instantaneous velocity of the object. Centripetal force cannot change the magnitude of the velocity, just the direction. Since velocity is a vector quantity, a change in direction is a change in velocity, Δv. The change in velocity Δv, can be found by subtracting the 2 vectors. (vectors are subtracted by placing them tail-to-tail). Δv = v2 – v1 Fc is always directed toward the center of the circle, as is the centripetal acceleration.

Centripetal acceleration ac = v2/r For magnitude of centripetal force Fc = mv2/r aka F=mac

Example: Centripetal Acceleration and Force A 0.25-kg mass is attached to a 1.00-m length of string. The mass completes a horizontal circle in 0.42 s. What is the velocity of the mass? What is the centripetal force acting on the mass?

Practice: p. 122 #12-13 HW: p. 122-123 #14-16

7:4 Placing a Satellite in Orbit Due to the shape and rotation of the Earth, an object with a horizontal speed of 8.00 x 103 m/s will remain the same altitude if air resistance is ignored. In space, there is very little friction, allowing satellites to orbit for long periods of time. The force of gravity on a satellite, its weight, provides the centripetal force to maintain its circular motion. Since F = ma and Fc = mv2/r, mg=mv2/r. Because of this, the velocity a satellite must have to orbit the earth is v = √(gr) Where g = acceleration of gravity at distance r.

A satellite farther from the earth has a larger velocity. The velocity of a satellite is independent of its mass A satellite must be given a large velocity to place it into orbit

HW practice: Pg. 124 #18-20

7:5 Simple Harmonic Motion An object with vibrational motion moves back and forth over the same path. Examples: A swinging pendulum and vibrating guitar strings. In simple harmonic motion, SHM, the motion of the object repeats a pattern. This repeated pattern is called a cycle. Time to complete one cycle is called a period, T. The number of cycles completed in a given time interval is called frequency, f. T = 1/f

SHM is a special vibrational motion that can best be described in terms of the motion of a mass on a string.

The motion of pendulums is another example of SHM. Amplitude of the swing is the distance from the equilibrium position, B, to the points of greatest displacements, A and C. The period, T, of a pendulum is the time interval for the bob to move from A to C and back to A.

Wavelength = Period

Gravitational acceleration of a pendulum can be resolved into 2 components. Fy is parallel to the string and opposes the force exerted by the string on the bob. Fx is perpendicular to the direction of the string. Represents the restoring force that accelerates the bob toward the equilibrium position.

The period of the simple pendulum of length l is expressed. The period of a pendulum depends on its length and the acceleration due to gravity, but not its mass. The period of a mass on a spring, however, depends on its mass and on the stiffness of the spring, but not on the acceleration of gravity.

The amplitude of any vibrating object can be greatly increased by applying a small external force at the same frequency as the vibrating object. This effect is called resonance. Example = “pumping”

Example: Suppose the period of the movement of a pendulum lasts 1.2 s on Earth. What is the length of the arm on the pendulum? Suppose this pendulum were placed on the surface of the moon, whose acceleration of gravity is 1.6 m/s2. What is the new period of the pendulum?

Chapter Review Pg. 130 #6 Pg. 130 problems A #1-8 Pg. 131 problems B #1-2