Lectures Page 1 LECTURE 18: Uniform Circular Motion (UCM) Select LEARNING OBJECTIVES: i. ii. iii. iv. v. vi. vii. viii. ix. x. xi. xii. xiii. xiv. xv. Understand the definition of UCM, specifically that the speed is constant but the tangential velocity is not constant, and that the radial acceleration is v2r while the tangential acceleration is zero. Be able to identify if a system is in uniform circular motion or not. Be able to draw a proper FBD with a well-chosen coordinate system. Be able to identify and label on a FBD the radial direction and tangential direction. Understand inertia related to circular motion. Demonstrate the ability to properly apply the mathematical expression for radial acceleration when using Newton's second law in conjunction with a FBD. Be able to identify any relevant angles on the FBD. Understand that centripetal force is a net force. Be able to identify which forces are responsible for UCM. Be able to identify direction of friction based off minimum or maximum constraints on speed. Be able to identify how forces are affected by minimum or maximum constraints on speed. Understand how the speed in UCM is related to the period of the object. Introduce the use of multiple view angles to help draw FBDs. Clarify the difference between centripetal force and the commonly heard, centrifugal force. TEXTBOOK CHAPTERS: Giancoli (Physics Principles with Applications 7 th ) :: 5-1, 5-2, 5-3 Knight (College Physics : A strategic approach 3 rd ) :: 6.1, 6.2, 6.3 BoxSand :: Forces ( Uniform Circular Motion ) WARM UP: A car is accelerating due east. What force is responsible for the car's acceleration? What direction does this force point?
Lectures Page 2 We would now like to analyze systems moving in a circular motion with a similar force analysis that we have been using thus far. A special case occurs when objects move in a circle with a constant speed. This case is called, uniform circular motion (UCM). Recall velocity has both a magnitude and direction, so even though the magnitude of the velocity (speed) is constant, the direction of the velocity is constantly changing. This is illustrated in the image below. Recall back in lecture 2 when we introduced plane polar coordinates, often abbreviated as polar coordinates. It was argued that this polar coordinate system is beneficial when certain symmetries exist. We never specified which symmetries at the time, but nevertheless every time we broke a Cartesian coordinate vector into a magnitude and direction, we were converting to polar coordinates. By definition, UCM describes an object moving around a circle with constant radius, at a constant speed. Note the image above. The symmetries of a circle allow us to specify the position vectors by a constant radius and an angle. Compare this to Cartesian coordinates where the x and y component are constantly changing as seen in the figure below.
Lectures Page 3 Notice how the x and y components of position constantly change as the object moves around the circle in the Cartesian coordinate system. Likewise, the x and y components of velocity change, and thus the acceleration components are also functions of time. This makes for a doable, yet complicated way to describe the motion of an object moving around in a circle at a constant speed. Now look at the polar coordinate system. The first thing to notice is that the coordinate system itself is changing as a function of time, always keeping the radial direction pointing along the radius of the current object position. Likewise, the tangential component is always tangent to the circle at the objects location on the circle. Exploiting this symmetry, the only time dependence we need to worry about is through the angular position, θ. Note that the velocity never has a radial component because the motion is around a circle which has a constant radius. Also note the acceleration does not have a tangential component because of our constraint of "uniform" motion (i.e. the object has a constant speed). **In the future, we will allow for objects to accelerate in the tangential direction when we get to rotational kinematics; for now we constrain ourselves to UCM, which by definition does not allow for a tangential acceleration. Also note, we could talk about a z-component for both the Cartesian and polar coordinates, which would point out of the page. This is orthogonal to the other two components in both coordinate systems, thus does not add any more complexity. By including a z-component, we could then call our coordinate system a cylindrical coordinate system. But for this lecture we will work with scenarios where the cicular motion lies in a plane which sets the z-component of everything equal to zero which is why we are not inlcuding it in our vector notation. To summarize, our position, velocity, and acceleration vectors for UCM look like Let's take a closer look at the acceleration during UCM by studying the image below. As seen above, the acceleration during UCM will always point towards the center of the circle. So the
Lectures Page 4 acceleration for UCM only has a component in the radial direction in polar coordinates. It can be shown that the magnitude of the radial acceleration is given by: Finally, we can now talk about using forces to analyze UCM. Luckily, there is not much different than what we have been doing before. We still choose a system, draw a free body diagram with all the forces acting on the system, but the key difference is to identify your coordinate system axes as the radial and tangential directions (z-direction if needed). This subtle difference shows up in the translation stage of using your FBD and Newton's 2 nd law to make a system of equations. Instead of setting the sum of the forces in each component to the linear acceleration in that component, we set the sum of the forces in the radial direction to the mass times the radial acceleration, and by definition of UCM, the sum of the forces in the tangential direction is zero. Let's continue with an example and many practice problems. EXAMPLE: A popular car magazine uses what is known as a circular "skid-pad" to determine the "lateral-g's" a car can withstand before skidding off the track. The skid-pad they use has a radius of roughly 103 ft, or 31.4 m. A 2016 Dodge Viper ARC tested on this track was reported to have a maximum lateral-g of 1.14g. What speed was the Viper going around the circular track before it started to skid? The 2016 Viper ACR has a reported mass of roughly 1530 kg. What was the magnitude of the force of friction responsible for making the Viper go around this circular track?
Lectures Page 5 PRACTICE: If an object moves in uniform circular motion, what can be said? (1) (2) (3) (4) (5) The velocity is constant. The speed is constant. The acceleration is constant. The net force is constant. The magnitude of displacement is constant. Picture a car driving around in a circle with a constant speed. This is an example of UCM. As you watch this car going round and round, notice that if you pick a point on the circle, the car will always wind up there after traveling around one full revolution. This repetitive motion is often described quantitatively by assigning what is known as the period (T), and frequency (f). The period (T) is the time it takes any quantity to complete one full cycle. In the car example, the quantity is its position and the cycle is one full revolution. Using the car analogy or any other UCM object, the period is defined as the time it takes an object to travel 1 full revolution. How many hours are in a day? The answer is 24 hours. This 24 hours is an example of a period. It takes the earth 24 hours to rotate once around its own axis. We can also quantify this repetitive motion by frequency (f). Frequency is defined as the number of cycles per a given unit of time (e.g. seconds). Going back to the car example or any object in UCM, frequency is defined as the number of revolutions an object completes in a given unit of time. Cars have speedometers which display the speed at which the car is going. They also have tachometers, which display frequency. For example, my Subaru has a redline of 6000 revolutions per minute (RPM). The tachometer is telling me the crank in my engine completes 6000 revolutions in 1 minute, which is a frequency. The period and frequency are really just two different approaches to quantify repetitive motion. Thus they are inately related to each other. Their relationship is PRACTICE: (a) What is the period of the second hand on a clock? (b) What is its frequency?
Lectures Page 6 PRACTICE: What is the speed of a point on the tip of a 3-cm-long second hand of a clock? (1) (2) (3) (4) (5) 1.57 m/s 1.57 x 10-3 m/s 6.28 x 10-4 m/s 3.14 x 10-5 m/s 3.14 x 10-3 m/s PRACTICE: A car travels around a circular corner of radius r at a constant speed. The corner is not banked. Which force is responsible for keeping the car on the road? 1. 2. 3. 4. 5. Normal Centripetal Friction Gravity Force of acceleration What is the maximum speed which the car can make this turn? PRACTICE: A ball is rolling on a horizontal table on the inside of a hula-hoop of radius r. At what speed must the ball be moving so that the normal force from the hoop is equal to the normal force from the floor?
Lectures Page 7 Assuming the ball is rolling counter-clock wise and is located at the dot on the figure below when the hoop is removed. Sketch the trajectory of the ball after the removal of the hoop. PRACTICE: A rollercoaster goes through a loop-the-loop of radius r. At the top of the loop (point A), what is the direction of the acceleration? At point A, what must the speed be to feel momentarily weightless? PRACTICE: A car is rolling over the top of a hill at speed v. At this instant, 1) The normal force is greater than the force of gravity
Lectures Page 8 PRACTICE: A car is rolling over the top of a hill at speed v. At this instant, 1) 2) 3) 4) The normal force is greater than the force of gravity The normal force is less than the force of gravity The normal force is equal to the force of gravity We can't tell how the normal force compares to the force of gravity without knowing v. PRACTICE: A car travels around a frictionless banked circular corner of radius r at a constant speed. Which force has a radial component? (1) (2) (3) (4) (5) Normal Centripetal Friction Gravity Force of acceleration
Lectures Page 9 PRACTICE: A car travels around a banked circular corner of radius r at a constant speed. Which direction does the force of static friction point if the car is going around the corner at the maximum speed possible before slipping? (1) (2) (3) (4) Up the incline Down the incline Forward Backwards Questions for discussion 1) 2) In the last practice problem with the car going around a banked turn with friction, why is there no centripetal force labeled in the FBD? When you "weigh" yourself, what does the scale read? What does weightless mean?