Final answer:
The difference in tension between the topmost and bottommost positions for a sphere performing vertical circular motion is calculated as 2mg by considering the centripetal force needed at each position. However, the correct answer of 40 N does not match any of the provided options, indicating an error in the question or the options.
Step-by-step explanation:
The question involves finding the difference in tension between the topmost and bottommost positions of a sphere performing vertical circular motion (VCM). To determine this, we need to consider the forces acting on the sphere at these two positions.
At the topmost position, the tension in the string, Ttop, plus the weight of the sphere must provide the centripetal force to keep it in circular motion. Therefore: Ttop + mg = m(v2/r), where m is the mass of the sphere, g is the acceleration due to gravity, v is the tangential velocity, and r is the radius of the circular motion.
At the bottommost position, the tension in the string, Tbottom, must overcome the weight of the sphere and provide the centripetal force. Therefore: Tbottom = m(v2/r) + mg.
The difference in tension between the bottommost and topmost positions is: ΔT = Tbottom - Ttop. When we calculate this by substituting mg with the given values, we end up with ΔT = 2mg.
Substituting the given mass (2 kg) and the acceleration due to gravity (10 m/s²), we get: ΔT = 2 * 2 kg * 10 m/s² = 40 N.
Since the question asks for the difference in tension between the topmost and bottommost positions, the correct answer is not listed in the options provided. As such, there must be an error in the question or the provided multiple-choice answers.