Angular velocity and centripetal acceleration



Angular velocity and centripetal accelerationHow can the motion of an object moving in a circle be measured and predicted? Is it accelerating? In what direction?ObjectivesMeasure the motion of an object moving in a circle.Use the measurements to develop the concept of angular velocity.Use the measurements to determine the relationship between angular velocity and centripetal acceleration.Materials and EquipmentData collection systemPASCO Rotating PlatformPASCO Wireless Acceleration Altimeter300-g square mass (included with rotating platform)SafetyFollow regular laboratory safety precautions.Procedure1.Level the rotating platform using the 300-g mass and instructions found in the rotating base manual.31648402020570Figure 1Figure 13164938224790002.Remove the 300-g mass and attach the wireless acceleration altimeter sensor to the end of the rotating platform arm using the screw and nut from the 300-g mass. Make sure the x-axis shown on the sensor is pointing toward the center of the rotating platform. See Figure 1.3.Measure the distance from the center of the rotating platform to the small circle with a dot on it on the case of the sensor. This distance is the radius from the center of rotation to the sensor. Record your value for the radius below.radius, r =m4.Connect the acceleration altimeter sensor to your data collection system and create an Angular Velocity–z versus Time graph display.5.Change the data collection rate to 50 Hz for both the gyro and acceleration sensor on the data collection software.6.Start data collection, then gently spin the platform and allow it to slow down. Spin it the other direction. Based on your observations, which direction for angular velocity is positive, clockwise or counterclockwise?7.Angular velocity is the rate of change of the angular position of the object. The angular velocity measured by the gyro in the sensor has units of radians/second (rad/s). How many radians are in one complete revolution of the accelerometer altimeter?8.Give the rotating platform a gentle counterclockwise push and then start recording data as the rotating bar passes a known location. Count how many times it rotates as it slows down to a stop, estimating to the nearest ?-rotation and record it below. Convert the number of rotations to radians. Each time it passes the start is an additional 2 radians. This represents the total angular displacement of the accelerometer altimeter. Show your work below.Number of rotations =9.The versus time graph display should be very close to linear. Assuming it is linear, the average angular velocity can be used to find the total angular displacement . The average angular velocity is the initial angular velocity 0 plus the final angular velocity divided by 2. The total angular displacement is the average angular velocity multiplied by the time. Calculate the average angular velocity and the total angular displacement using the values from the angular velocity graph. Show your work below.pare the calculation for total angular displacement with the estimate from counting the number of revolutions. Calculate the percent error assuming the value found from the average angular velocity is the theoretical value. Show all of your work below.11.There are other ways to calculate the total angular displacement. The area-under-the-curve method works if the angular velocity graph is not linear. Use the tools of your data collection software to find the area under the angular velocity versus time graph. This represents the total angular displacement. Compare it to the value you calculated using the average angular velocity above. Calculate the percent error assuming the value found from the area is the actual value.12.If the screw that attaches the sensor to the rotating arm is removed, the sensor would not stay in place as it spins. This implies there is a force needed to keep it attached. This force causes an acceleration directed toward the center of the rotating arm. In Latin, "toward the center," translates to, "centripetal." This centripetal acceleration ac changes the direction of the sensor’s velocity as it moves in a circular path. Predict what will happen to ac as increases.13.Test your prediction by creating a graph display with Acceleration–x (ac) on the vertical axis and Angular Velocity–z () on the horizontal axis using the data collection software. Give the rotating platform a gentle counterclockwise push and then start recording data. Stop recording after it comes to a stop. Describe the shape of your graph. Explain below how the graph supports or refutes your prediction.14.The graph of ac versus was not linear. Its shape suggests a parabola where one of the quantities varies with the square of the other. This can be tested by modifying the graphed data. Use the tools of the data collection software to create a graph of ac versus 2 and/or ac2 versus to find which one results in a linear graph. Describe what you learned from the graph(s) below.15.Use the tools of the data collection software to find the slope of the graph of ac versus 2. What are the units of the slope? Remember that radians are dimensionless, so they have no units. The value of the slope should be very close to another quantity measured in this lab that has the same units. What quantity is it? Record the value of your slope and show your work for finding its units below.16.Because the ac versus 2 graph was linear, we can easily find an equation relating the two variables. Use y = mx + b to develop this equation. Replace y with the variable graphed on the vertical axis, ac. Replace x with the variable graphed on the horizontal axis, 2. You should have noticed that the radius value you measured previously was very close to the value for your slope, and the units are both in meters. This is strong evidence that the slope m can be replaced by r. The y-intercept b is the acceleration when the sensor is not rotating. Write the new equation for centripetal acceleration below.17.Test the assumption that the slope of the ac versus 2 graph represents the radius by moving the sensor to a new location. Loosen the screw and slide it about half-way toward the center. Measure the new radius and record it below. Give the rotating platform a gentle counterclockwise push and then start recording data. Stop recording after it comes to a stop. Use the tools of the data collection software to find the slope of the graph of ac versus 2. Compare the slope to the measured value of r. Find the percent error assuming the measured value is the actual value. Show all of your work below.radius, r =m18.The equation for centripetal acceleration ac is often written in terms of the speed v instead of the angular velocity . When an object is moving in a circular arc, the distance it travels x is the angular displacement , measured in radians, multiplied by the radius. Speed v is distance x divided by the time. Angular velocity is the angular displacement divided by the time. Use this information to determine the relationship between speed and angular velocity below. Justify your result.19.The speed of an object moving in a circle is the angular velocity multiplied by the radius, v = r. Use this relationship and the equation developed for ac to write an equation for ac in terms of v and r. Show all of your work below.20.Assume there are two sensors attached to the rotating platform. Number 1 is on the end of the rotating platform and number 2 is attached about half-way in. The platform is given a counterclockwise push. At any given time while the platform is rotating, which sensor, 1 or 2, will measure the greatest angular velocity? Which sensor will measure the greatest speed? Which sensor will measure the greatest centripetal acceleration?21.A tight circular curve on a road has an inside and an outside lane for cars going in the same direction. Two cars are taking the turn at the same time at the same speed, one on the inside lane and one on the outside. Which will experience the greatest acceleration? Explain your reasoning using the equations for ac.22.Which would result in a linear graph, a graph of ac versus v2 or a graph of ac2 versus v? For the choice that results in a linear graph, what would be the significance of its slope?23.Long-term exposure to the free fall condition of an orbiting space station has serious negative health consequences for astronauts. One idea that would help is to build a space station in the shape of a wheel and rotate it. The astronaut's feet would be located on the inner surface of the wheel's rim with their heads pointing toward the center of the space station. Because their velocity must be constantly changing direction, the floor would push on them, causing a centripetal acceleration. This push from the floor must be directed toward the center. The space station could be spun fast enough so that the centripetal acceleration was the same as the free-fall acceleration on Earth. Other than seeing the floor curve up in front of them while looking down a long hallway, they may not notice any difference between this experience and standing on Earth. Assume the space station will not be rotated faster than once per minute so the astronauts don't get sick. What radius would it need to have so that the centripetal acceleration is 9.81 m/s2? Why do you think a space station like this has not been built yet? Show all of your work and use correct units. ................
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