Question

19.55 The 8-kg uniform bar AB is hinged at C and is attached at A to a spring of constant k 5 500 N/m. If end A is given a small displace- ment and released, determine


(a) the frequency of small oscillations,


(b) the smallest value of the spring constant k for which oscillations will occur.

Answer 1

Answer:

Explanation:

C is center of mass of the bar ie middle point. Spring is attached with one end ie A of the bar . When this end is displaced by distance x ( small) a restoring force kx is produced which creates a torque

Torque created = kx . l /2  , which creates angular acceleration α

moment of inertia I = ml²/12

torque = I. x α

kx . L /2 = I. x α

α = L kx / 2I

a = linear acceleration

= α x L/2

=  L² kx / 4I

= L²x 12 kx / 4 mL²

a  = 3kx/m

This shows that motion is SHM as acceleration is proportional to displacement x .

angular frequency ω² = 3k / m

frequency f = 1/2π √ 3k/m

= (1/6.28) x √ 3x 500/8

= 2.18 Hz