Circular Motion (College Board AP® Physics 1: Algebra-Based): Exam Questions

A small object of mass is attached to a string of variable length and set into conical pendulum motion, where the object moves in a horizontal circular path while the string traces out a cone. The system allows for adjustment of the string length and measurement of the angular velocity of the object.

A group of students is asked to verify the relationship between the angular velocity and the angle .

i) Draw a labeled diagram of the experimental setup

ii) List the equipment needed to measure the required quantities.

Describe an experimental procedure for varying and measuring the corresponding .

Describe how the data can be analysed to confirm the theoretical relationship.

Discuss two potential sources of error in your experiment and how they could affect the results.

A satellite of mass orbits a planet of mass in a circular orbit at an altitude above the planet’s surface. The radius of the planet is .

A Physicist states that the satellite orbits the planet obeying the relationship:

Sketch on Figure 1 a graph of period as a function of orbital speed for the satellite in its circular orbit around the planet.

Figure 1

A ball of mass is attached to a string and swings in a vertical circle of radius . At the lowest point of the circle, the tension in the string is measured to be .

Starting with Newton's second law, derive an expression for the speed of the ball at the lowest point in terms of , and any physical constants that are appropriate.

Derive an expression for the minimum speed required at the top of the circle for the ball to complete the motion without the string going slack. Express your answer in terms of , and any physical constants as appropriate.

A car of mass travels around a banked curve of radius slower than the ideal speed of for the given banking angle. The coefficient of static friction between the tires and the road is to be determined.

Starting from Newton's second law, derive an expression for the ideal banking angle required for the car to stay on the curve without friction at the ideal speed.

The car is moving slower than the ideal speed at a speed of . Derive an expression for the minimum coefficient of static friction required to prevent sliding, and determine the direction in which it will slide.

The speed of the car is increased beyond the ideal value. Indicate whether the direction of the frictional force acts up the incline, down the incline, or perpendicular to it. Justify your reasoning.

A car travels around a banked curve of radius at a speed of . The banking angle is given as .

On high-speed highways, ideal banking angles are typically around for speeds of about .

Estimate whether the ideal banking angle is less than, the same as, or greater than the given banking angle. Justify your answer quantitively.

Starting from Newton's second law, derive an expression for in terms of .

"If the coefficient of friction were sufficiently high, the car could still navigate the curve safely at despite the banking angle being lower than the ideal angle."

Do you agree with this statement? Justify your answer.

A small sphere of mass is attached to a string of length and is swung such that it moves in a vertical circle. At the highest point, the speed of the sphere is just enough to maintain circular motion. The string is then adjusted so that the sphere moves in a horizontal conical pendulum motion with the same speed.

A group of students have a protractor with a vertical plumb line, a ruler, a high speed camera and a stopwatch. The students are asked to compare the relationship between the length of the string and the tensions in the strings for the vertical circle and the conical pendulum motion.

Describe an experimental procedure the student could use to collect data that would allow them to compare the tensions in the strings at different lengths. Include any steps necessary to reduce experimental uncertainty. If needed, you may include a simple diagram of the setup with your procedure.

Describe how the collected data should be plotted to create a linear graph and how that graph would be analyzed to compare the tensions in the strings in both experiments.

Figure 1 shows a small sphere of mass attached to a light string of length and is swung in a vertical circle.

At the lowest point, the sphere encounters a small peg located a distance above the lowest point, causing it to transition into a smaller circular motion about the peg.

Figure 1

Starting with the equation for centripetal acceleration, derive an expression for the force exerted by the string as a function of speed at the lowest point of its orbit before contact with the peg is made.

The sphere is just on the point of maintaining circular motion at the top of the smaller circular path. Derive an expression for the reaction force exerted by the peg on the string at this point.

After the string catches onto the peg, the sphere now moves in a smaller circular path of radius . Identify whether the sphere maintains circular motion, and justify your reasoning using energy conservation principles.