# dynamics and inertia

The vehicle s used to transport supplies to and from the bottom of the 25-present grade. Each par of wheels one at A and the other at B has a mass of 140 kg with a radius of gyration of 100 mm. The total mass of the vehicle s 520 g. The vehicle is released from rest with a restraining force T of 500 N n the control cable which passes around the drum and is secured at D. The wheels roll without slipping.

a) Calculate the inertia of wheel A about its centre in kg.m^2

b) Calculate the inertia of wheel C about its centre in kg.m^2

c) Determine the initial acceleration a of the vehicle in m/s^2

Refer to attachment for diagram

Â© BrainMass Inc. brainmass.com December 24, 2021, 11:21 pm ad1c9bdddfhttps://brainmass.com/physics/applied-physics/dynamics-inertia-562122

## SOLUTION This solution is **FREE** courtesy of BrainMass!

a)

mass of A, mA = 140 kg

Radius of gyration, kA = 150 mm == 0.15 m

inertia I = m*k^2

inertia of A,

IA = mA * kA^2 = 140*0.15^2 = 3.15 kg.m^2

b)

mass of C, mC= 40 kg

radius of gyration of C, kC = 100 mm = 0.1 m

Hence, inertia of C,

IC = mC*kC^2 = 40*0.1^2 = 0.4 kg.m^2

c)

Vehicle moving downward.

Let us assume, acceleration of the vehicle = a

Therefore, cable acceleration = 2*a

Hence, drum C angular acceleration:

alphaC = 2*a/rC

rC (radius of C) = 150 mm = 0.15 m

Let us assume tension in other side of the cable (towards D) = T2

Hence

(T2 - T)*rC = IC*alphaC = IC *2*a/rC

=> T2 = T + 2*IC*a/rC^2

For wheel A, and B, let us assume friction forces are fA and fB acting upward -- supporting rotation of wheels.

Angular acceleration of A (alphaA) = angular acceleration of B (alphaB) = a/rA

rA = rB = 200 mm = 0.2 m

Hence,

fA*rA = IA*alphaA

=> fA = IA*a/rA^2

similarly,

fB = IB*a/rB^2

IA = IB = 3.15 kg.m^2

Hence,

fA = fB = IA*a/rA^2 = IB*a/rB^2

Now, for complete system of the vehicle,

mass of the system, M = 520 kg

T = 500 N

M*g*sin(x) - T - T2 - fA - fB = M*a

where, x: angle of the plane with horizontal.

tan(x) = 25/100 = 1/4 == sin(x) (For small angle).

=> M*g*sin(x) - T - (T + 2*IC*a/rC^2) - IA*a/rA^2 - IA*a/rA^2 = M*a

=> M*g*sin(x) - 2*T= (M + 2*IC/rC^2 + 2*IA/rA^2)*a

=> a = (M*g*sin(x) - 2*T)/(M + 2*IC/rC^2 + 2*IA/rA^2)

=> a = (520*9.8*1/4 - 2*500)/(520 + 2*0.4/0.15^2 + 2*3.15/0.2^2) = 0.38 m/s^2

Â© BrainMass Inc. brainmass.com December 24, 2021, 11:21 pm ad1c9bdddf>https://brainmass.com/physics/applied-physics/dynamics-inertia-562122