Question

A child is playing in a park on a rotating cylinder of radius, r , is set in rotation at an angular speed of w as shown below. The base of the cylinder is slowly moved away, leasing the child suspended against the wall in a vertical position. What is the minimum coefficient of friction between the child's clothing and wall is needed to prevent it from falling.

Answer 1

Complete Question:

A child is playing in a park on a rotating cylinder of radius, r = 3.00 m , is set in rotation at an angular speed of w = 5.00 rad/s. The base of the cylinder is slowly moved away, leaving the child suspended against the wall in a vertical position. What is the minimum coefficient of friction between the child's clothing and wall needed to prevent it from falling

Answer:

minimum coefficient of friction between the child's clothing and wall needed to prevent it from falling,
`\mu = 0.131`

Step-by-step explanation:

Applying the Newton's law:

`\mu N = mg`...............(1)

Where N = Normal reaction.

and
`\mu` = coefficient of friction

Since the cylinder is a rotating one, the normal reaction will be calculated using the formula
`N = (mv^(2) )/(r)`.................(2)

Substituting (2) into (1)

`\mu ( mv^(2) )/(r) = mg`.............(3)

v = wr..........(4)

Substitute (4) into (3)

`\mu ( m\omega^(2) *r^(2) )/(r) = mg\\\mu \omega^(2) *r = g\\\mu = (g)/(\omega^(2) r )`

Substituting, w, g, and r into the equation above

Angular speed,
`w = 5 rad/s`

Radius, r = 3 m

g = 9.8 m/s²

`\mu = (9.8)/(5^(2) *3 )`

`\mu = 9.8/75\\\mu = 0.131`