Calculate the maximum displacement that can be realized when you are allowed – [Free] B37
Calculate the maximum displacement that can be realized when you are allowed to move along the surface of a sphere. Please write your answer as a function of the radius of the sphere.
Physics Problem – Maximum Displacement on a Sphere
Step 1: Understand what is meant by “maximum displacement”.
We are asked to find the maximum straight-line distance (also known as displacement) from the starting point when movement is restricted to the surface of a sphere.
Step 2: Consider geometry on a sphere.
On a sphere, the maximum possible displacement occurs when a person moves from a point to its antipodal point — the point that lies exactly on the opposite side of the sphere.
The two points form a straight line (a chord) that goes through the center of the sphere.
Step 3: Use the chord length formula.
Let r be the radius of the sphere and θ the central angle between the starting and ending points. The straight-line distance (chord length) between two points on a sphere is:
The maximum value of sin(θ / 2) is 1, which happens when θ = π radians (i.e., the angle between antipodal points).
Step 4: Calculate maximum displacement.
Thus, the maximum displacement is equal to twice the radius of the sphere.