Question

A circular hoop (?hoop ?? ) of mass ? 5.00 kg, radius ? 2.00 m, and infinitesimal thickness rolls without slipping down a ramp inclined at an angle ? 32.0° with the horizontal.

(a) Draw a free body diagram of the hoop. [1]

(b) Determine the magnitude of the hoop’s acceleration down the ramp. [4]

(c) Determine the force of static friction on the hoop. [3]

Answer 1

(a)

b)

a = linear acceleration of the hoop parallel to incline

= angular acceleration of the hoop parallel to incline

r = radius of the hoop = 2 m

m = mass of hoop = 5 kg

moment of inertia of the hoop is given as

I = m r²

I = (5) (2)²

I = 20 kgm²

weight of the hoop is given as

F_g = mg

F_g = 5 x 9.8 = 49 N

force equation parallel to incline is given as

F_g Sin - f_s = ma

f_s = F_g Sin - ma eq-1

Torque equation for the hoop is given as

=  f_s r = I

f_s r = I

f_s r = I (a/r) since = a /r

f_s = I (a/r²)

using eq-1

F_g Sin - ma = I (a/r²)

inserting the values

(49) Sin32 - 5a = 20 (a/(2)²)

a = 2.6 m/s²

c)

using eq-1

f_s = F_g Sin - ma

inserting the values

f_s = 49 Sin32 - (5 x 2.6)  

f_s = 13 N