Question

A child lies on his back and raises his head up off the floor. When doing so, the total tension force in his neck muscles is 51.0 N. The same child now is sliding feet first down a water slide. The slide makes a circular curve, where the outside wall of the slide is vertical but the curve itself is in the horizontal plane, and the radius of the curve is 2.40 m. While sliding along this curve, the child's speed is 3.35 m/s. The child raises his head from the wall of the slide and holds it steady, looking forward. At this moment, what is the total tension (in N) in the child's neck muscles

Answer 1

The total tension in the child's neck muscle, T = 56.51 N

Step-by-step explanation:

Let m = mass of the child's neck

Radius of the curve, r = 2.40 m

The child's speed, v = 3.35 m/s

The tension force on the child's neck when he raises his head up off the floor,
`T_(f) = 51.0 N`

The tension force on the child's neck when he raises his head from the wall of the slide,
`T_(s) = ?`

`T_(f) = mg\\g = 9.8 m/s^2\\51 = m * 9.8\\m = 51/9.8\\m = 5.2 kg`

Since he makes a circular turn in water, the radial acceleration can be given by the equation:

`a_(r) = (v^(2) )/(r) \\a_(r) = (3.35^(2) )/(2.4)\\a_(r) = 4.68 m/s^2`

`T_(s) = ma_(r) \\T_(s) = 5.2 * 4.68\\T_(s) = 24.336 N`

The total tension in the child's neck muscle till be calculated as:

`T = \sqrt{T_(f) ^(2) + T_(s) ^(2) } \\T = \sqrt{51 ^(2) + 24.336^(2) }\\T = √(2601 + 592.24 )\\T = 56.51 N`