Question

A bob of mass 4kg is tied to the roof by a rope of length
4m. If the bob and rope rotate as a conical pendulum
(bob moves on a horizontal circle) with semi vertex angle
0 = 30° then, find the net force acting on the bob. Take g =
10.​

Answer 1

Given : A bob of mass 4kg is tied to the roof by a rope of length 4m. If the bob and rope rotate as a conical pendulum

(bob moves on a horizontal circle) with semi vertex angle, θ = 30°.

To find : the net force acting on the bob.

solution : here r = lsin30° = 4m × 1/2 = 2m

now, time period of conical pendulum is given by, T = 2π√{lcosθ/g}

= 2π √{4 × √3/2/10}

= 2π√{√3/5}

= 2π × 0.6

= 1.2π

Now angular frequency, ω = 2π/T = 2π/1.2π = 5/3 rad/s

at equilibrium,

Tcos30° = mg .......(1)

Tsin30° = mω²r .......(2)

from equations (1) and (2) we get,

T = √{(mg)² + (mω²r)}

= √{(4 × 10)² + (4 × 25/9 × 2)²}

= √{1600 + (200/9)²}

= √(1600 + 493.7284)

= 45.75 N

therefore the net force acting on the bob is 45.75 N (approx)