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. A group of students is asked to verify the relationship between the angular velocity of the object and the angle the string makes with the vertical.

Describe an experimental procedure the student could use to collect data that would allow them to verify the relationship between the angular velocity of the object and the angle of the string. 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 analyzed to verify the relationship between the angular velocity of the object and the angle of the string.

A small object of mass is attached to a string of variable length and set into a 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.

If the length of the string is increased, indicate whether the linear velocity of the mass increases, decreases, or remains constant. Justify your reasoning.

A student is investigating the relationship between the length of a simple pendulum and its period of oscillation. The student has access to a stopwatch, a meter rule, a pendulum bob, a retort stand with a clamp, and a protractor.

Describe an experimental procedure the student could use to collect data to verify the relationship between the length of the pendulum and the period of oscillation. 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 verify the relationship between length and period.

The experiment is now modified to investigate the forces acting on a mass moving in a vertical circle. The student releases a pendulum bob from a horizontal position and measures the tension in the string at the lowest point of the swing. The student also measures the velocity of the bob at this point. The recorded data is shown in the table.

The relationship between tension, velocity, and the length of the string is given by:

Where:

• is the tension in the string

• is the mass of the pendulum bob

• is the velocity at the lowest point

• is the length of the string,

• 

i) Indicate two quantities that could be plotted to yield a linear graph whose slope could be used to determine the length of the string. Use the blank columns in the table to list any calculated quantities you will graph other than the data provided.

Vertical Axis: ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ Horizontal Axis: ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

ii) Plot on the grid in Figure 2 the data points for the quantities indicated in part (c)(i). Clearly scale and label all axes, including units.

Figure 2

iii) Draw a best-fit line through the data graphed in part (c)(ii).

iv) Determine the experimental value of the gradient of the best-fit line.

Calculate the length of the string using your experimental value of the slope from part (c)(iv)

Figure 1

A spacecraft of mass is in a clockwise circular orbit of radius around Earth, as shown in Figure 1. The mass of Earth is .

Express your answers for i) and ii) in terms of , , and physical constants as appropriate. In each case, begin your derivation by writing a fundamental physics principle or an equation from the reference information.

i) Derive an equation for the orbital speed of the spacecraft.

ii) Derive an equation for the orbital period of the spacecraft.

A second spacecraft of mass is placed in a circular orbit with the same radius .

Is the orbital period of the second spacecraft greater than, less than, or equal to the orbital period of the first spacecraft? Briefly justify your reasoning.

Figure 1

A ball of mass is attached to a rope of length and swings with constant speed in a vertical circle, as shown in Figure 1. At Point A, the ball passes the lowest point in its path, and at Point B, it makes an angle of with the horizontal. The magnitude of the tension in the rope at Point B is equal to three-quarters of the magnitude of the tension in the rope at Point A.

Starting with Newton's second law, derive an expression for the tension in the rope as the ball passes Point A. Express your answer in terms of , , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Determine an expression for the speed of the ball in terms of and 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 of mass travels around a banked curve of radius at a constant speed of . The coefficient of static friction between the surface and the tyres is . The first section of the banking makes an angle with the horizontal. In this section, no frictional force acts on the car. The second section of the banking makes an angle with the horizontal. In this section, the frictional force only just prevents the car from sliding up the banking.

Indicate whether is greater than, equal to, or less than .

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Justify your answer using qualitative reasoning beyond referencing equations.

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.