A pendulum made of a string and a sphere is able to swing in a vertical plane. The pendulum is released from a position of 60° from vertical as shown in the figure below.
The string hits a peg located a distance d below the point of suspension and rotates about the peg.
The acceleration of gravity is 9.8 m/s?

Find the smallest value of d (highest peg position) in order for the sphere to swing in a full circle centered on the peg.
Answer in units of m.

To find the smallest value of \( d \) (the highest peg position) for the pendulum to complete a full circle, we'll use energy conservation and some basic kinematic principles.

### Step 1: Define the problem

- The pendulum consists of a mass \( m \) hanging from a string of length \( L \).
- It is released from a position where the string makes an angle of \( \theta = 60^\circ \) with the vertical.
- The pendulum swings down, and at some point, the string hits a peg located a distance \( d \) below the point of suspension.
- After hitting the peg, the pendulum rotates around the peg.

To complete a full circle around the peg, the velocity of the sphere when it reaches the top of the circular motion around the peg must be non-zero. We need to find the smallest value of \( d \) for which this condition is met.

### Step 2: Conservation of Energy

When the pendulum is released, its potential energy at the highest point is converted into kinetic energy as it swings down. Let's calculate the energies:

- **Initial Potential Energy** at release (at \( \theta = 60^\circ \)):  
  The height of the mass above the lowest point is given by:  
  \[
  h_1 = L(1 - \cos \theta) = L(1 - \cos 60^\circ) = L(1 - 0.5) = 0.5L
  \]
  So the initial potential energy is:  
  \[
  PE_1 = mgh_1 = mg \cdot 0.5L
  \]

- **Kinetic Energy** at the point just before it hits the peg (since it starts from rest, the total mechanical energy is conserved):  
  At this point, the height is \( d \) above the lowest point. So the potential energy at this point is:
  \[
  PE_2 = mgd
  \]
  The kinetic energy is then the difference between the initial potential energy and the potential energy at the height of the peg:
  \[
  KE_2 = PE_1 - PE_2 = mg(0.5L - d)
  \]

- **Energy at the top of the swing around the peg**:  
  For the pendulum to complete a full circle around the peg, it needs to have just enough kinetic energy at the top of the circle to ensure non-zero speed. The height at the top of the circle is \( 2(L - d) \) (the diameter of the circle), so the potential energy at the top is:
  \[
  PE_3 = mg \cdot 2(L - d)
  \]
  The kinetic energy at the top must be at least zero, meaning:
  \[
  KE_3 = 0
  \]
  Thus, the total mechanical energy at the top of the circle is all potential energy:
  \[
  E_3 = mg \cdot 2(L - d)
  \]

### Step 3: Applying Energy Conservation

Since energy is conserved from the point of release to the top of the circular path, we equate the energy at the point just before it hits the peg to the energy at the top of the circle:
\[
mg(0.5L - d) = mg \cdot 2(L - d)
\]

### Step 4: Solve for \( d \)

Canceling out \( mg \) from both sides and simplifying:
\[
0.5L - d = 2(L - d)
\]
Expanding:
\[
0.5L - d = 2L - 2d
\]
Rearranging terms to isolate \( d \):
\[
0.5L + d = 2L
\]
\[
d = 2L - 0.5L
\]
\[
d = 1.5L
\]

Thus, the smallest value of \( d \) for the pendulum to complete a full circle around the peg is \( d = 1.5L \).

### Conclusion:

The highest position of the peg that allows the pendulum to complete a full circle is \( d = 1.5L \), where \( L \) is the length of the string.

A pendulum made of a string and a sphere is able to swing in a vertical plane. The pendulum is released from a position of 60° from vertical. The string hits a peg located a distance d below the point of suspension and rotates about the peg. The acceleration of gravity is 9.8 m/s². Find the smallest value of d (highest peg position) in order for the sphere to swing in a full circle centered on the peg. Answer in units of m.

Learn how to calculate the smallest value of d (the peg height) for a pendulum to complete a full circular swing around the peg. This problem involves energy conservation principles, including potential and kinetic energy, and is suitable for students in Grades 11-12.

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