A 30 kg child on swing 2m long is released from rest
when the swing supports make an angle of 37 with the
vertical
a) neglecting friction, find the child's speed at lowest position
b) if the speed of the lowest position is 2.2m/s, what is the energy loss
due to friction?
a) Using conservation of energy, we can find the child's speed at the lowest position:
Initial energy = Potential energy at highest point
1/2(mv^2) + mgh = mgh
1/2(v^2) = g(2 - 2cos37)
v^2 = 19.6(2 - 2cos37)
v = 4.19 m/s
Therefore, the child's speed at the lowest position is 4.19 m/s.
b) To find the energy loss due to friction, we can use the conservation of energy again:
Initial energy = Final energy
1/2(mv^2) + mgh = 1/2(mv_l^2) + mgh_l + E_loss
where v_l is the speed at the lowest position and h_l is the height at the lowest position.
Solving for E_loss:
E_loss = 1/2(mv^2 - mv_l^2) - mg(h - h_l)
E_loss = 1/2(30)(4.19^2 - 2.2^2) - 9.8(2 - 2cos37)
E_loss = 39.7 J
Therefore, the energy loss due to friction is 39.7 J.
To find the child's speed at the lowest position on the swing, we can use the principle of energy conservation. We'll assume that the child starts from rest and neglect friction.
a) Let's calculate the child's speed at the lowest point.
1. Determine the potential energy at the highest point (initial position).
At the highest point, all the potential energy is converted into gravitational potential energy.
Potential energy = mass × gravity × height
Potential energy = 30 kg × 9.8 m/s² × 2 m
Potential energy = 588 J
2. Determine the kinetic energy at the lowest point.
At the lowest point, all the potential energy is converted into kinetic energy.
Kinetic energy = 0.5 × mass × velocity²
588 J = 0.5 × 30 kg × velocity²
velocity² = 588 J / (0.5 × 30 kg)
velocity² = 19.6 m²/s²
velocity = √(19.6 m²/s²)
velocity ≈ 4.43 m/s
Therefore, neglecting friction, the child's speed at the lowest position is approximately 4.43 m/s.
b) To find the energy loss due to friction, we can subtract the final energy (after friction) from the initial energy.
1. Determine the final kinetic energy at the lowest point.
Final kinetic energy = 0.5 × mass × velocity²
Final kinetic energy = 0.5 × 30 kg × (2.2 m/s)²
Final kinetic energy ≈ 72.6 J
2. Determine the energy loss due to friction.
Energy loss = Initial energy - Final energy
Energy loss = 588 J - 72.6 J
Energy loss ≈ 515.4 J
Therefore, if the speed at the lowest position is 2.2 m/s, the energy loss due to friction is approximately 515.4 J.