University Physics: Mechanics Lecture 6 Ch4. TWO- AND THREE-DIMENSIONAL MOTION University Physics: Mechanics Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com 2015

Uniform Circular Motion A particle is in uniform circular motion if it travels around a circle or a circular arc at constant (uniform) speed. Although the speed does not vary, the particle is accelerating because the velocity changes in direction. The velocity is always directed tangent to the circle in the direction of motion. The acceleration is always directed radially inward. Because of this, the acceleration associated with uniform circular motion is called a centripetal (“center seeking”) acceleration.

Uniform Circular Motion → The magnitude of this centripetal acceleration a is: (centripetal acceleration) where r is the radius of the circle and v is the speed of the particle. In addition, during this acceleration at constant speed, the particle travels the circumference of the circle (a distance of 2πr) in time of: (period) with T is called the period of revolution, or simply the period, of the motion.

Centripetal Acceleration

Checkpoint An object moves at constant speed along a circular path in a horizontal xy plane, with the center at the origin. When the object is at x = –2 m, its velocity is –(4 m/s) j. Give the object’s (a) velocity and (b) acceleration at y = 2 m. ^ v2 → a → 2 m v1 → v1 = –4 m/s j ^ → v2 = –4 m/s i ^ → a = –8 m/s2 j ^ →

Example: Fighter Pilot Fighter pilots have long worried about taking a turn too tightly. As a pilot’s body undergoes centripetal acceleration, with the head toward the center of curvature, the blood pressure in the brain decreases, leading to unconsciousness. What is the magnitude of the acceleration, in g units, of a pilot whose aircraft enters a horizontal circular turn with a velocity of vi = 400i + 500j m/s and 24 s later leaves the turn with a velocity of vf = –400i – 500j m/s? → ^ ^ → ^ ^

Example: Aston Martin An Aston Martin V8 Vantage has a “lateral acceleration” of 0.96g. This represents the maximum centripetal acceleration that the car can attain without skidding out of the circular path. If the car is traveling at a constant speed of 144 km/h, what is the minimum radius of curve it can negotiate? (Assume that the curve is unbanked.) The required turning radius r is proportional to the square of the speed v Reducing v by small amount can make r substantially smaller

Relative Motion in One Dimension The velocity of a particle depends on the reference frame of whoever is observing or measuring the velocity. For our purposes, a reference frame is the physical object to which we attach our coordinate system. In every day life, that object is the ground.

Thom(p)son Encounters Relative Velocity

Relative Motion in One Dimension Suppose that Alex (at the origin of frame A) is parked by the side of a highway, watching car P (the ”particle”) speed past. Barbara (at the origin of frame B) is driving along the highway at constant speed and is also watching car P. Suppose that both Alex and Barbara measure the position of the car at a given moment. From the figure we see that “The coordinate of P as measured by A is equal to the coordinate of P as measured by B plus the coordinate of B as measured by A”

Relative Motion in One Dimension Taking the time derivative of the previous equation, we obtain “The velocity of P as measured by A is equal to the velocity of P as measured by B plus the velocity of B as measured by A”

Relative Motion in One Dimension Here we consider only frames that move at constant velocity relative to each other. In our example, this means that Barbara drives always at constant velocity vBA relative to Alex. Car P (the moving particle), however, can accelerate. Constant

Example: Relative Velocity Suppose that Barbara’s velocity relative to Alex is a constant vBA = 52 km/h and car P is moving in the negative direction of the x axis. (a) If Alex measures a constant vPA = –78 km/h for car P, what velocity vPB will Barbara measure?

Example: Relative Velocity Suppose that Barbara’s velocity relative to Alex is a constant vBA = 52 km/h and car P is moving in the negative direction of the x axis. (b) If car P brakes to a stop relative to Alex (and thus relative to the ground) in time t = 10 s at constant acceleration, what is its acceleration aPA relative to Alex? (c) What is the acceleration aPB of car P relative to Barbara during the braking?

Relative Motion in Two Dimensions In this case, our two observers are again watching a moving particle P from the origins of reference frames A and B, while B moves at a constant velocity vBA relative to A. The corresponding axes of these two frames remain parallel, as shown, for a certain instant during the motion, in the next figure. → The following equations describe the position, velocity, and acceleration vectors:

Example: Sail Through the River A boat with the maximum velocity of 5 m/s aims to cross the river from F to H. H is located directly on the other side of the river, 20 m to the east of F. The speed of current is 1.5 m/s due south. F H 1.5 m/s 20 m Determine how the boat driver should direct the boat so that it can sail due east directly from F to H; The sailor should direct the boat 17.46° north of due east

Example: Sail Through the River A boat with the maximum velocity of 5 m/s aims to cross the river from F to H. H is located directly on the other side of the river, 20 m to the east of F. The speed of current is 1.5 m/s due south. F H 1.5 m/s 20 m (b) Calculate the time of trip from F to H. The time of trip in presence of current is somehow greater than the one when the boat sails in still water, (20 m) / (5 m/s) = 4 s.

Trivia: Frog Crossing the River A certain species of frog has a unique characteristic. Every time the frog jump forward for 3.0 m, it will jump backward for 1.0 m. The frog never jumps forward twice in a row. If the frog must cross a river 17.0 m wide, how many jumps does it need? Solution: Only one jump. Afterwards the frog will swim until it reaches the other side of the river!

Example: Plane Moves West A plane moves due east while the pilot points the plane somewhat south of east, toward a steady wind that blows to the northeast. The plane has velocity vPW relative to the wind, with an airspeed (speed relative to wind) of 215.0 km/h, directed at angle θ south of east. The wind has velocity vWG relative to the ground with speed 65.0 km/h, directed 20.0° east of north. What is the magnitude of the velocity vPG of the plane relative to the ground, and what is θ. → → →

Example: Plane Moves West

Exercise Problems A cat rides a mini merry-go-round turning with uniform circular motion. At time t1 = 2 s, the cat’s velocity is v1 = 3i + 4j m/s, measured on a horizontal xy coordinate system. At t2 = 5 s, its velocity is v2 = –3i – 4j m/s. What are (a) the magnitude of the cat’s centripetal acceleration and (b) the cats average acceleration during the time interval t2 – t1? ^ ^ Answer: (a) 5.236 m/s2; (b) –2i – 2.667 j m/s2. A suspicious-looking man runs as fast as he can along a moving sidewalk from one end to the other, taking 2.50 s. Then security agents appear, and the man runs as fast as he can back along the sidewalk to his starting point, taking 10.0 s. What is the ratio of the man’s running speed to the sidewalk’s speed? Answer: 1.67