Lab_8_Circ Motion - PHYS 181 FA23 001-2 004-10 013-15 017

Circular Motion When an object is accelerating its velocity must be changing. However, this does not mean that its speed is changing. If we consider an object travelling around in a circle at uniform speed, then it must always be accelerating even though its speed is constant. This is because the direction of its velocity is constantly changing. Newton’s second law relates the force on an object to its acceleration. So, this means that there must be a net force that causes an object to go in a circle. If there is no net force, the object will travel in a straight path. Consider the case of a ball at the end of a string that is being twirled in a horizontal circle on a table at uniform speed. In order for the ball to travel in a circle, there must be a constant acceleration pointing towards the center of the circle. This is known as the centripetal acceleration and it always points radially inward. Newton’s second law states that the net force must be equal to the mass of the object times its acceleration. So, in this case mv? Foet =ma, = - 1) where a, is the centripetal acceleration, v is the speed of the ball and r is the radius of the circle. We see that the faster the ball is moving the larger the centripetal acceleration. Similarly, the smaller the circle the larger the acceleration must be because the direction must be changing more quickly. In the case of the ball, the net force is provided by the tension in the rope. If the rope was suddenly cut, the ball would fly off in an initially straight path. In the first part of the lab, you will be measuring the force required to keep different masses rotating at constant speeds. In the second part of the lab, you will be examining the motion of a roller coaster when it is upside down. In this case, the roller coaster must be moving fast enough that the centripetal acceleration is larger than the acceleration due to gravity. You will find the minimum height that the roller coaster must start from in order to complete a loop without falling off the track. Goals of this lab: o Understand the concept of centripetal acceleration. e Measure the force required for uniform circular motion. e Determine the minimum height for a roller coaster to complete a loop. 78

Circular Motion Lab equipment: Rotating arm — The rotating arm is spun by hand. There is a rotary :;__-’_'H’—-—" 5 motion sensor on the bottom and two masses on it. Force sensor This force sensor can measure forces up to 50 N. It will be used to measure the force on the rotating mass. Rotary motion sensor This rotary motion sensor measures the angular velocity of the rotating arm. Roller Coaster The car will travel along the roller coaster track. Car This car can travel on the roller coaster path. The flag is 4.7 mm wide and 10 mm between the two arms. Photogates The photogates have an infrared sensor in them that will be used to measure the velocity of the car with the Sparklink interface. Sparklink The Sparklink interface connects to a computer by USB and uses the Capstone software to acquire data from up to 4 sensors at once. 79

Circular Motion Set up the rotating arm as shown in the picture above. Load the lab in the PASCO Capstone software and switch to the tab to measure the force versus rotational velocity. For instructions on setting up the Capstone software, refer to the appendix of the lab manual. e Pick a mass of around 200 grams and choose a position near the middle of the arm. You can adjust the position of the mass by moving the stand with the force sensor up or down. " Measure the mass of the object and its distance from the center. Don’t forget to measure the mass of the black slider and the additional mass. Record these values into the table on the worksheet. ® Make sure that the rotating arm is well balanced by moving the counter weight on the other side. When it is properly balanced, the arm should not turn by itself when tilted. Press the zero button on the force sensor. Start turning the arm until it is rotating at its maximum speed which should be around w = 10 rad/s. " Record data while the arm slows to a stop. " Use the coordinates tool to measure the force and angular velocity. Make sure that you are displaying at least 4 digits for each. You should record the force and angular velocity at equal intervals of angular velocity in the table on the worksheet. You should have at least 10 different values of the angular velocity. II. Force versus mass In this part of the lab, you will be varying the mass and recording the tension for a fixed radius. You will need to take all of the measurements at the same angular velocity because as you saw in the first section, the angular velocity also changes the force. The setup is exactly the same as the previous part. e Pick a convenient angular velocity to use for the measurement and choose a position near the middle of the arm for the mass. ® Measure the distance from the center. Record this value and the angular velocity that you will use into the table on the worksheet. * You are going to use five different masses in this part of the lab. When you put each mass on, you need to make sure that the rotating arm is well balanced by moving the counter weight on the other side. When it is properly balanced, the arm should not turn by itself when tilted. Press the zero button on the force sensor. Start turning the arm until it is rotating faster than the angular velocity that you are using for your measurements. " Record data while the arm slows to a stop. " After the arm has stopped rotating, use the coordinates tool to measure the force and angular velocity. Make sure that you are displaying at least 4 digits for each. You should find the entry with the angular velocity that you are using. Then record the force at this angular velocity in the table on the worksheet. Repeat this measurement with the other masses until you have measured 5 different masses. 81

Circular Motion IIL. Force versus radius In this part of the lab, you will be varying the distance of the mass and recording the tension. You will need to take all of the measurements with the same mass and at the same angular velocity because as you saw in the previous two parts, the angular velocity and mass also changes the force. The setup is exactly the same as the previous parts. o Pick a convenient angular velocity to use for the measurement and choose a mass around 200 grams. ® Measure the mass. Record this value and the angular velocity that you will use into the table on the worksheet. * You are going to use five different distances in this part of the lab. You adjust the distance by moving the stand with the force sensor up or down. After doing this, you need to make sure that the rotating arm is well balanced by moving the counter weight on the other side. When it is properly balanced, the arm should not turn by itself when tilted. Press the zero button on the force sensor. Start turning the arm until it is rotating faster than the angular velocity that you are using for your measurements. " Record data while the arm slows to a stop. " After the arm has stopped rotating, use the coordinates tool to measure the force and angular velocity. Make sure that you are displaying at least 4 digits for each. You should find the entry with the angular velocity that you are using. Then record the force at this angular velocity in the table on the worksheet. Repeat this measurement with the other distances until you have measured 5 different distances. These distances should span the range from the central axis to the far end of the rotating arm. IV. Minimum height for roller coaster loop In an earlier lab, we saw that the roller coaster car can make it around the loop without falling off the track. Now, we are going to figure out how high the car must start from in order to stay on the track all the way around the loop. The setup for this portion of the lab is shown in the picture below. [ 1 [=15) Photogate

Name: Date: Circular Motion Worksheet I. Force versus rotation speed You should fill out the table below (don’t forget the units). Mass: Radius: Angular Velocity Force You should make a plot of the force as a function of the angular velocity squared. This graph should be labeled GRAPH#1. Fit a straight line to the data in GRAPH#1. The slope should be equal to mr. Measured value of the slope. Calculated value of the slope from measurement of mass and radius. II. Force versus mass You should fill out the table below (don’t forget the units). Angular Velocity: Radius: Mass Force 84

Worksheet: Circular Motion You should make a plot of the force as a function of the mass. This graph should be labeled GRAPH#2. Fit a straight line to the data in GRAPH#2. The slope should be equal to rw?. Measured value of the slope. Calculated value of the slope from measurement of radius and angular velocity. III. Force versus radius You should fill out the table below (don’t forget the units). Mass: Angular Velocity: Radius Force You should make a plot of the force as a function of the radius. This graph should be labeled GRAPH#3. Fit a straight line to the data in GRAPH#3. The slope should be equal to mw?2. Measured value of the slope. Calculated value from the measurement of the mass and angular velocity. IV. Minimum height for roller coaster loop Measured minimum height for roller coaster car. Measured minimum speed at top of loop. ‘What changes when the mass of the car is increased? 85