A bob is whirled in a horizontal plane by means of a string with an initial speed of ω rpm. The tension in the string is T. If speed becomes 2ω while keeping the same radius, the tension in the string becomes:
When an object is whirled in a horizontal plane, the tension in the string provides the centripetal force required to keep the object in circular motion. The centripetal force is given by the formula:
F = m ⋅ ω² ⋅ r
where m is the mass of the object, ω is the angular velocity, and r is the radius of the circular path.
Initially, the tension T in the string is equal to the centripetal force:
T = m ⋅ ω² ⋅ r
When the speed becomes 2ω, the new tension T' in the string is:
T' = m ⋅ (2ω)² ⋅ r
Simplifying, we get:
T' = m ⋅ 4ω² ⋅ r
Therefore, the new tension T' is four times the initial tension T:
T' = 4T
Step-by-Step Solution:
Step 1: Identify the initial tension in the string
T = m ⋅ ω² ⋅ r
Step 2: Determine the new angular velocity:
2ω
Step 3: Calculate the new tension in the string using the new angular velocity:
T' = m ⋅ (2ω)² ⋅ r
Step 4: Simplify the expression to find the new tension:
