A child of mass m sits in a swing of negligible mass suspended by a rope of length l. Assume that the dimensions of the child are negligible compared with l.
His father pulls the child back until the rope makes an angle of one radian with vertical, then pushes with a force F=mg along the arc of the circle releasing at the vertical :
a) How high up will the swing go?
b) How long did the father push?

To tackle this problem, we need to analyze the motion of the swing and the forces acting on the child. The scenario involves some fundamental concepts from physics, particularly energy conservation and circular motion. Let's break it down step by step.

Understanding the Swing's Motion

When the father pulls the child back to an angle of one radian, the swing is at a certain height above its lowest point. When he pushes the child with a force equal to the child's weight along the arc, he provides the child with kinetic energy that will allow the swing to rise again after being released. We can use the principles of energy conservation to find out how high the swing will go after the push.

Part A: Determining the Height

Initially, when the swing is pulled back to an angle of one radian, we can calculate the height (h) above the lowest point using the formula:

• h = l - l * cos(θ)

Here, θ is the angle in radians (1 radian), and l is the length of the rope. Substituting the values, we get:

• h = l - l * cos(1)
• h = l(1 - cos(1))

Next, when the father pushes the child with a force F = mg, he does work on the child. The work done (W) by the father can be expressed as:

• W = F * d

Where d is the distance along the arc that the child moves while being pushed. Since the force is applied along the arc, we can consider the arc length corresponding to the angle of one radian:

• d = l * θ = l * 1 = l

Thus, the work done by the father is:

• W = mg * l

Now, we can apply the principle of conservation of energy. The total mechanical energy at the lowest point (just after the push) will equal the potential energy at the highest point:

• mg(h + l) = mg(h_max)

Where h_max is the maximum height reached after the push. Rearranging gives us:

• h_max = h + l = l(1 - cos(1)) + l = l(2 - cos(1))

Part B: Calculating the Duration of the Push

To find out how long the father pushed, we need to consider the dynamics of the swing. The force exerted by the father is equal to the weight of the child, which means that the swing is accelerated along the arc. The acceleration (a) can be calculated using Newton's second law:

• a = F/m = mg/m = g

Since the swing moves along a circular path, we can relate the linear acceleration to the angular acceleration (α) using the radius (l):

• a = l * α

Thus, we have:

• g = l * α
• α = g/l

The angular displacement (θ) during the push is 1 radian. The time (t) taken to push can be calculated using the kinematic equation for rotational motion:

• θ = α * t

Substituting the values gives us:

• 1 = (g/l) * t
• t = l/g

Final Results

In summary, the maximum height the swing will reach after the push is:

• h_max = l(2 - cos(1))

And the duration of the push is:

• t = l/g

This analysis illustrates how energy conservation and basic dynamics can be applied to understand the motion of a swing in a straightforward manner. If you have any further questions or need clarification on any part, feel free to ask!