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 apply some principles of physics, particularly energy conservation and kinematics. 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 (F = mg), he is essentially providing energy to the system. Once released, the swing will convert this energy into potential energy as it rises to a new height.

Calculating the Initial Height

First, we need to find out how high the swing is when pulled back to an angle of one radian. The height (h) can be calculated using the formula:

• h = l - l * cos(θ)

Here, θ is the angle in radians, and l is the length of the rope. Substituting θ = 1 radian:

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

Using the approximate value of cos(1) ≈ 0.5403, we find:

• h ≈ l(1 - 0.5403) ≈ l(0.4597)

Energy Conservation to Find Maximum Height

When the father pushes the swing, he does work on the system. The work done by the father is equal to the force times the distance along the arc. Since the force is equal to the weight of the child (mg), the work done (W) can be expressed as:

• W = F * d = mg * d

Here, d is the distance along the arc from the initial position to the vertical position. This distance can be calculated as:

• d = l * θ = l * 1 = l

Thus, the work done by the father is:

• W = mg * l

Now, at the lowest point, all the energy (potential + kinetic) will convert into potential energy at the maximum height (h_max). The potential energy at the maximum height is given by:

• PE = mgh_max

Setting the work done equal to the potential energy at the maximum height:

• mg * l = mgh_max

We can cancel m from both sides (assuming m is not zero), leading to:

• g * l = gh_max

From this, we find:

• h_max = l

Determining the Duration of the Push

To find out how long the father pushed the swing, we need to consider the dynamics of the swing. The swing moves along a circular path, and the force applied by the father acts tangentially to this path. The acceleration (a) of the swing can be calculated using Newton's second law:

• a = F/m = mg/m = g

Since the swing is moving along a circular path, the tangential acceleration is equal to g. The time (t) taken to push the swing can be calculated using the kinematic equation:

• d = 0.5 * a * t^2

Substituting d = l and a = g:

• l = 0.5 * g * t^2

Rearranging gives us:

• t^2 = (2l)/g
• t = sqrt(2l/g)

Final Results

In summary, the swing will rise to a height of:

• h_max = l

And the time the father pushed the swing is:

• t = sqrt(2l/g}

This analysis shows how energy conservation and kinematics work together to describe the motion of the swing effectively. If you have any further questions or need clarification on any part, feel free to ask!