Intro to Center of Mass Practice Problems

A circle is made by bending a rod having a uniform mass distribution. It's then cut in half along the diameter. If the radius of the circle was r, find the center of mass of the semicircle for a coordinate system that has its origin at the center of the circle.

Calculate the distance from the nitrogen atom to the center of mass of a nitrogen monoxide (NO) molecule, where the nitrogen atom (m = 14 u) and the oxygen atom (m = 16 u) are separated by 1.14 x 10⁻¹⁰ m.

On a nearly frictionless ice surface, a 53-kg skater and a 73-kg hockey player stand 10.1 m apart. Calculate how far from the skater their center of mass is.

The axis of a cone is along the x-axis, while the apex of the cone is at the origin. The distance from the apex to the base of the cone is H and the radius of the base is r. Determine the center of mass of the cone assuming it has a uniform mass distribution.

[Hint: Consider the cone being made up of an infinite number of thin cylinders of thickness dx as shown in the figure.]

Determine the position of the center of mass of a circular table, relative to the center of the table, weighing 7.1 kg and having a radius of 35 cm, which is unbalanced due to a 1.51-kg decorative piece placed 25 cm from the table's center.

Consider a 1251-kg unoccupied van. Initially, the center of mass of the van is located 2.41 m behind the front. Now, consider the scenario where one individual is seated in the front seat, positioned 2.81 m from the front, and two individuals are seated in the back seat, which is located 3.91 m from the front. Determine the distance of the new center of mass from the front of the van. Assume each person weighs 66 kg.

Calculate the center of mass of a phosphine molecule (PH₃) relative to the center of its base. The molecule consists of a phosphorus atom at the apex of a pyramid and three hydrogen atoms at the corners of an equilateral triangle that forms the base. The side of the triangle is 0.14 nm, and the vertical distance from the phosphorus atom to the plane of hydrogen atoms is 0.10 nm.

Determine the distance from the Jupiter-Ganymede system's center of mass to Jupiter, given that Jupiter has a mass of 1.90 x 10²⁷ kg and Ganymede has a mass of 1.48 x 10²³ kg, with their centers separated by 1.07 x 10⁶ m.

A research balloon, equipped for atmospheric studies, is stationary at a high altitude above the Earth's surface. It has a total mass $M_1$ , including both the balloon and attached scientific instruments. For specific data collection, one instrument package, with a mass $M_2$ , needs to be lowered to a different altitude. This instrument package is allowed to slide down a rope at a speed v relative to the balloon. Calculate the velocity $V_b$ of the balloon relative to the Earth immediately after the package starts sliding. What happens to the balloon's velocity relative to the Earth if the instrument stops sliding?