A child with a weight of 230 N swings on a playground swing attached to 1.90 m long chains. What is the gravitational potential energy (in J) of the child-earth system relative to the child's lowest position at the following times?

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Question:

A child with a weight of 230 N swings on a playground swing attached to 1.90 m long chains. What is the gravitational potential energy (in J) of the child-earth system relative to the child's lowest position at the following times?

(a)when the ropes are horizontal

(b) when the ropes make a 25.0° angle with the vertical  

(c) when the child is at the bottom of the circular arc

Given Information:  

Weight = W = 230 N

Length of rope = L = 1.90 m

Required Information:  

a) Gravitational potential energy when the ropes are horizontal  = ?

(b) Gravitational potential energy when the ropes make a 25.0° angle with the vertical   = ?

(c) Gravitational potential energy when the child is at the bottom of the circular arc

a)   GPE = 437 J

(b)  GPE = 41.07 J

(c)   GPE = 0 J

Explanation:

As we know, gravitational potential energy is given by

GPE = mgh

Where mg = W and h = L(1 - cosθ)

a) Gravitational potential energy when the ropes are horizontal  = ?

In this case, the ropes are perpendicular with respect to vertical therefore, the angle θ is 90°

GPE = 230*1.90(1 - cos(90°))

GPE = 230*1.90(1 - 0)

GPE = 437 J

(b) Gravitational potential energy when the ropes make a 25.0° angle with the vertical   = ?

In this case, the ropes are at 25° with respect to vertical therefore, the angle θ is 25°

GPE = 230*1.90(1 - cos(25°))

GPE = 437(1 - 0.906)

GPE = 41.07 J

(c) Gravitational potential energy when the child is at the bottom of the circular arc

In this case, the ropes are aligned with respect to vertical therefore, the angle θ is 0°

GPE = 230*1.90(1 - cos(0°))

GPE = 437(1 - 1)

GPE = 437(0)

GPE = 0 J

d is the correct answer