A small object with mass m is tied to one end of a string and is being swung in a horizontal circle of radius r with a constant speed v. The other end of the string is attached to a fixed point.
(a) State the formula for the centripetal force acting on the object in terms of m, v, and r.
(b) If the mass of the object is doubled while keeping the speed and radius constant, how does the centripetal force change? Justify your answer mathematically.
(a) The formula for the centripetal force F acting on the object in terms of m, v, and r is given by the equation:
F = (m * v^2) / r
Where:
F is the centripetal force acting on the object (in newtons, N)m is the mass of the object (in kilograms, kg)v is the speed of the object (in meters per second, m/s)r is the radius of the circular path (in meters, m)(b) To determine how the centripetal force changes when the mass is doubled while keeping the speed and radius constant, we substitute 2m for m in the formula and compare the resulting forces. Let's calculate the new force:
F_new = (2m * v^2) / r
To compare F_new with F, we calculate the ratio:
F_new / F = ((2m * v^2) / r) / ((m * v^2) / r)
= (2m * v^2 * r) / (m * v^2 * r)
= 2
The ratio F_new / F is equal to 2, which means that when the mass is doubled while keeping the speed and radius constant, the centripetal force also doubles. This can be justified mathematically by cancelling out the common factors m, v^2, and r in the equation.
Therefore, the centripetal force increases directly proportionally to the mass of the object in this scenario.