A child bounces in a harness suspended from a door frame by three elastic bands.

(a) If each elastic band stretches 0.210 m while supporting a 7.35-kg child, what is the force constant for each elastic band?


(b) What is the time for one complete bounce of this child?


(c) What is the child's maximum velocity if the amplitude of her bounce is 0.210 m?

To find the force constant for each elastic band, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. The formula for Hooke's Law is given by:

F = k * x

Where:
F is the force exerted by the spring
k is the force constant (spring constant)
x is the displacement of the spring

(a) The given information states that each elastic band stretches 0.210 m while supporting a 7.35-kg child. We can use this information to find the force constant for each elastic band.

Using Hooke's Law:
F = k * x
k = F / x

The weight of the child can be calculated using the formula:
F = m * g
Where:
m is the mass of the child = 7.35 kg
g is the acceleration due to gravity = 9.8 m/s^2

Substituting the values into the equation:
F = (7.35 kg) * (9.8 m/s^2)
F = 71.73 N

Using this value of force and given displacement:
k = (71.73 N) / (0.210 m)
k ≈ 341.57 N/m

Therefore, the force constant for each elastic band is approximately 341.57 N/m.

(b) To find the time for one complete bounce of the child, we need to consider the period of oscillation. The period of oscillation is the time taken for one complete cycle. It can be calculated using the formula:

T = 2π * √(m / k)

Where:
m is the mass of the child = 7.35 kg
k is the force constant = 341.57 N/m

Substituting the values into the equation:
T = 2π * √(7.35 kg / 341.57 N/m)
T ≈ 2π * √0.0215
T ≈ 2π * 0.1469
T ≈ 0.92 s

Therefore, the time for one complete bounce of the child is approximately 0.92 seconds.

(c) The maximum velocity of the child can be calculated using the formula:

v_max = A * ω

Where:
A is the amplitude of the bounce = 0.210 m
ω is the angular frequency, which can be calculated as ω = √(k / m)

Substituting the values into the equation:
ω = √(341.57 N/m / 7.35 kg)
ω ≈ √46.53
ω ≈ 6.82 rad/s

v_max = (0.210 m) * (6.82 rad/s)
v_max ≈ 1.43 m/s

Therefore, the child's maximum velocity is approximately 1.43 m/s.

(a) To find the force constant for each elastic band, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. The equation for Hooke's Law is:

F = kx

Where F is the force applied, k is the force constant, and x is the displacement.

In this case, each elastic band stretches 0.210 m while supporting a 7.35-kg child. We can use this information to find the force constant (k) for each elastic band.

F = kx
k = F / x

Given:
x = 0.210 m
F = mg (where g is the acceleration due to gravity and m is the mass of the child)

Plugging in the values:
k = (mg) / x = (7.35 kg)(9.8 m/s^2) / 0.210 m

Calculate the force constant (k) using the above equation.

(b) To find the time for one complete bounce of the child, we can use the concept of simple harmonic motion (SHM). The formula to calculate the time period (T) of SHM is:

T = 2π * √(m / k)

Where m is the mass and k is the force constant.

Given:
m = 7.35 kg (mass of the child)
k = (calculate from part a)

Plugging in the values:
T = 2π * √(7.35 kg / k)

Calculate the time period (T) using the above equation.

(c) The maximum velocity of the child can be found using the equation for simple harmonic motion:

v_max = A * ω

Where v_max is the maximum velocity, A is the amplitude of the bounce, and ω is the angular frequency.

The angular frequency (ω) can be calculated using the formula:

ω = √(k / m)

Given:
A = 0.210 m (amplitude of the bounce)
k = (calculate from part a)
m = 7.35 kg (mass of the child)

Plugging in the values:
ω = √(k / m)
v_max = A * ω

Calculate the maximum velocity (v_max) using the above equations.

a) hookes law

b) c) isn't this simple harmonic motion, and there are standard formulas for this?