A thin uniform rod of length 2 m, cross-sectional area A and density d is rotated about an axis passing through its centre and perpendicular to its length with angular velocity ω. If ω = √(αE/Ad), where E is its rotational kinetic energy, then the value of α is ______.
Physics · Circular Motion
JEE Mains 30 January 2023 Shift 1 Morning Shift

Question

A thin uniform rod of length 2 m, cross-sectional area A and density d is rotated about an axis passing through its centre and perpendicular to its length with angular velocity ω. If ω = √(αE/Ad), where E is its rotational kinetic energy, then the value of α is ______.

Physics · Circular Motion

JEE Mains 30 January 2023 Shift 1 Morning Shift

G1 Physics · Accepted Answer

Mass m = density×volume = dA(2). Moment of inertia about the centre is I = mL²/12 = [2dA×(2)²]/12 = (2/3)dA. Rotational energy E = (1/2)Iω² = (1/3)dAω². Hence ω² = 3E/(Ad), so α = 3.