The problem involves a pendulum which is a simple harmonic motion problem. We can use the conservation of energy to solve this problem.

Step 1: Determine the initial potential energy (PE) of the pendulum.

The height h1 at which the pendulum bob is initially released can be calculated using the formula h1 = L - Lcosθ, where L is the length of the pendulum and θ is the angle. Substituting the given values, we get h1 = 1 - cos60° = 1 - 1/2 = 1/2 m.

The initial potential energy (PE1) is then given by mgh1, where m is the mass of the bob (which we'll later see cancels out), g is the acceleration due to gravity, and h1 is the initial height. Substituting the given values, we get PE1 = mg(1/2) = 1/2 mg.

Step 2: Determine the final potential energy (PE) of the pendulum.

The height h2 at which we want to find the rate of change of speed can be calculated similarly using the formula h2 = L - Lcosθ. Substituting the given values, we get h2 = 1 - cos30° = 1 - √3/2 ≈ 0.134 m.

The final potential energy (PE2) is then given by mgh2. Substituting the given values, we get PE2 = mg(0.134) = 0.134 mg.

Step 3: Use the conservation of energy to find the final kinetic energy (KE).

The total mechanical energy of the system is conserved, so the initial potential energy (PE1) equals the final potential energy (PE2) plus the final kinetic energy (KE). Therefore, KE = PE1 - PE2 = 1/2 mg - 0.134 mg = 0.366 mg.

Step 4: Use the kinetic energy to find the final speed of the pendulum bob.

The kinetic energy (KE) is given by 1/2 mv², where m is the mass of the bob and v is its speed. Therefore, v = √(2KE/m) = √(2*0.366g) = √7.32 ≈ 2.7 m/s.

Step 5: Use the speed to find the rate of change of speed.

The rate of change of speed is the derivative of speed with respect to time. However, without a given time, we cannot directly calculate this. But we can say that the speed of the pendulum bob is increasing as it swings down from 60° to 30°.

Question

The problem involves a pendulum which is a simple harmonic motion problem. We can use the conservation of energy to solve this problem.

Step 1: Determine the initial potential energy (PE) of the pendulum.

The height h1 at which the pendulum bob is initially released can be calculated using the formula h1 = L - Lcosθ, where L is the length of the pendulum and θ is the angle. Substituting the given values, we get h1 = 1 - cos60° = 1 - 1/2 = 1/2 m.

The initial potential energy (PE1) is then given by mgh1, where m is the mass of the bob (which we'll later see cancels out), g is the acceleration due to gravity, and h1 is the initial height. Substituting the given values, we get PE1 = mg(1/2) = 1/2 mg.

Step 2: Determine the final potential energy (PE) of the pendulum.

The height h2 at which we want to find the rate of change of speed can be calculated similarly using the formula h2 = L - Lcosθ. Substituting the given values, we get h2 = 1 - cos30° = 1 - √3/2 ≈ 0.134 m.

The final potential energy (PE2) is then given by mgh2. Substituting the given values, we get PE2 = mg(0.134) = 0.134 mg.

Step 3: Use the conservation of energy to find the final kinetic energy (KE).

The total mechanical energy of the system is conserved, so the initial potential energy (PE1) equals the final potential energy (PE2) plus the final kinetic energy (KE). Therefore, KE = PE1 - PE2 = 1/2 mg - 0.134 mg = 0.366 mg.

Step 4: Use the kinetic energy to find the final speed of the pendulum bob.

The kinetic energy (KE) is given by 1/2 mv², where m is the mass of the bob and v is its speed. Therefore, v = √(2KE/m) = √(2*0.366g) = √7.32 ≈ 2.7 m/s.

Step 5: Use the speed to find the rate of change of speed.

The rate of change of speed is the derivative of speed with respect to time. However, without a given time, we cannot directly calculate this. But we can say that the speed of the pendulum bob is increasing as it swings down from 60° to 30°.

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