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 a child swinging from a height after being pushed, and we can break it down into two parts: the height the swing reaches and the duration of the push. Let’s dive into each part step by step.

Determining the Maximum Height of the Swing

When the father pulls the swing back to an angle of one radian, the child is lifted to a certain height above the lowest point of the swing. To find out how high the swing goes after being released, we can use the principles of energy conservation.

Energy Conservation Principle

Initially, when the swing is pulled back, the child has gravitational potential energy (PE) due to the height gained. When the swing is released and reaches the lowest point, this potential energy is converted into kinetic energy (KE). At the highest point of the swing after being pushed, all the kinetic energy will again convert back into potential energy.

Calculating the Initial Height

When the swing is pulled back to an angle θ (in this case, 1 radian), the vertical height (h) can be calculated using the formula:

• h = l - l * cos(θ)

Substituting θ = 1 radian:

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

Now, when the swing is released, the potential energy at height h is:

• PE_initial = mgh = mg * l(1 - cos(1))

After the push, the swing will convert this potential energy into kinetic energy at the lowest point and then back into potential energy at the maximum height (H) it reaches after the push:

• PE_final = mgH

Setting the initial potential energy equal to the final potential energy gives us:

• mg * l(1 - cos(1)) = mgH

We can cancel out mg from both sides (assuming m is not zero):

• l(1 - cos(1)) = H

Thus, the maximum height (H) the swing reaches after being pushed is:

• H = l(1 - cos(1))

Calculating the Duration of the Push

Next, we need to determine how long the father pushed the swing. The force applied by the father (F = mg) acts along the arc of the swing. To find the time of the push, we can analyze the motion of the swing during that period.

Understanding the Motion

When the father pushes the swing, he applies a force that accelerates the child along the circular path. The swing moves from the angle of 1 radian to the vertical position. The distance along the arc (s) can be calculated as:

• s = l * θ

For θ = 1 radian:

• s = l * 1 = l

Finding the Acceleration

The net force acting on the child when the father pushes is the gravitational force minus the tension in the rope. However, since the father is pushing with a force equal to mg, the net force is effectively:

• F_net = F - mg = 0

This means the child experiences uniform circular motion, and we can use the formula for the time taken (t) to travel a distance s under constant acceleration:

• t = √(2s/a)

Here, 'a' is the centripetal acceleration, which can be derived from the gravitational force acting on the child. However, since the push is instantaneous and we assume the swing moves smoothly, we can simplify this by considering the average speed during the push.

Using the average speed (v) during the push, we can estimate the time:

• t = s/v

Assuming the swing reaches maximum speed at the lowest point, we can use the formula for the speed at the lowest point derived from energy conservation:

• v = √(2gH)

Substituting H from earlier gives us:

• v = √(2g * l(1 - cos(1)))

Finally, substituting back into the time equation:

• t = l / √(2g * l(1 - cos(1)))

This provides a clear method to calculate how long the father pushed the swing. By analyzing the forces and energy transformations, we can understand the dynamics of the swing effectively.