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

A small object of mass $m$ is attached to a string of variable length $L$ 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 $\omega$ of the object and the angle $\theta$ the string makes with the vertical.

i) Indicate quantities that could be measured by the students that would allow them to verify the relationship between the angular velocity of the object and the angle of the string, using a linear graph.

ii) Briefly describe a method to reduce experimental uncertainty for the measured quantities.

i) Indicate what quantities students could plot on the horizontal and vertical axes to create a linear graph that can be used to verify the relationship between the angular velocity of the object and the angle of the string.

ii) Briefly describe how the graph could be analyzed to verify the predicted relationship $\omega =\sqrt{\frac{g}{L\text{cos}\theta }}$.

A small object of mass $m$ is attached to a string of variable length $L$ 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 $L$ and measurement of the angular velocity $\omega$ 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 investigates the forces acting on the pendulum bob as it moves in a vertical circle. The student uses a pendulum bob of mass $m=0.15\mathrm{kg}$ and a string of unknown length $L$. A force sensor is used to measure the tension $T$ in the string, and a motion sensor is used to measure the speed $v$ of the bob as it passes through the lowest point of the swing. The recorded data is shown in Table 1.

Table 1

The student correctly determines that the relationship between $T$ and $v$ at the lowest point is given by:

$T-mg=\frac{m{v}^2}{L}$

The student wants to determine an experimental value for the length $L$ of the string.

i) Label the axes of Figure 1 with measured or calculated quantities. Include units, as appropriate. The graphed quantities should yield a linear graph that can be used to determine $L$.

ii) On the grid provided in Figure 1, create a graph of the quantities indicated in part a)i).

• Clearly label the horizontal and vertical axes with numerical scales.

• Plot the corresponding data points on the grid

• Any columns added to Table 1 for scratch work will not be scored.

Figure 1

iii) Draw a best fit line for the data plotted in part a)ii).

Calculate an experimental value for the length of the string using the best-fit line that you drew in part a)iii).

Figure 1

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

Express your answers for i) and ii) in terms of $m$, $M_E$, $R$ 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 $v$ of the spacecraft.

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

A second spacecraft of mass $2m$ is placed in a circular orbit with the same radius $R$.

Indicate whether the orbital period of the second spacecraft is greater than, less than, or equal to the orbital period of the first spacecraft by writing one of the following:

• $T_2>T_1$

• $T_2=T_1$

• $T_2<T_1$

Justify your reasoning.

Figure 1

A ball of mass $m$ is attached to a rope of length $L$ and swings with constant speed $v$ 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 $30°$ 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 $v$, $m$, $L$, 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 $v$ of the ball in terms of $L$ and physical constants as appropriate.

A car of mass $m$ travels around a banked curve of radius $r$ at a speed slower than the ideal speed of $v_{ideal}$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 at speed $v$which is slower than the ideal speed.

Derive an expression for the minimum coefficient of static friction required to prevent sliding.

The speed of the car is increased beyond the ideal value.

Indicate the direction of the frictional force acting on the car by writing one of the following:

• up the incline

• down the incline

• perpendicular to the incline

Justify your reasoning.