An 85 cm long pendulum with a 0.80 kg bob is released from rest at an initial angle of θ0 with the vertical. At the bottom of the swing, the speed of the bob is 2.8 m/s.

What angle does the pendulum make with the vertical when the speed of the bob is 1.4 m/s?

dude, you do it

Dmw

To find the angle that the pendulum makes with the vertical when the speed of the bob is 1.4 m/s, we can use the principle of conservation of mechanical energy. The mechanical energy of the system remains constant throughout the motion.

The total mechanical energy of the pendulum is the sum of its kinetic energy and potential energy. At the highest point of the swing, when the bob is momentarily at rest, all of the energy is in the form of potential energy. At the lowest point of the swing, when the bob has maximum speed, all of the energy is in the form of kinetic energy.

Let's denote the initial angle with the vertical as θ0 and the angle we want to find as θ.

At the bottom of the swing, the speed of the bob is given as 2.8 m/s. Using the equation for kinetic energy:

Kinetic energy = (1/2) * mass * (speed)^2

Substituting in the known values:

(1/2) * (0.80 kg) * (2.8 m/s)^2 = (1/2) * (0.80 kg) * (1.4 m/s)^2

Now we can solve for θ.

To do this, we can use the conservation of mechanical energy, which states that the total mechanical energy of the system remains constant. In this case, the total mechanical energy is the sum of the potential and kinetic energies.

At the highest point of the swing, when the bob is momentarily at rest, all of the energy is in the form of potential energy. At the lowest point of the swing, when the bob has maximum speed, all of the energy is in the form of kinetic energy.

The potential energy of the pendulum can be calculated using the equation:

Potential energy = mass * gravitational acceleration * height

The height of the bob can be found using trigonometry. At the highest point of the swing, the height is given by:

Height = length of pendulum * (1 - cos(θ))

At the lowest point of the swing, the height is zero, since the bob is at its lowest position.

The potential energy at the highest point is equal to the kinetic energy at the lowest point:

mass * gravitational acceleration * height = (1/2) * mass * (speed)^2

Substituting the previously calculated values, we can solve for θ:

(0.80 kg) * (9.8 m/s^2) * [length of pendulum * (1 - cos(θ))] = (1/2) * (0.80 kg) * (1.4 m/s)^2

Simplifying the equation, we can solve for θ:

length of pendulum * (1 - cos(θ)) = (1/2) * (1.4 m/s)^2 / (9.8 m/s^2)

Now, we can rearrange the equation to solve for θ:

cos(θ) = 1 - [(1/2) * (1.4 m/s)^2 / (9.8 m/s^2 * length of pendulum)]

Finally, we can use the inverse cosine function to find the value of θ:

θ = cos^(-1) [1 - [(1/2) * (1.4 m/s)^2 / (9.8 m/s^2 * length of pendulum)]]

Substituting the given values of length of pendulum (85 cm = 0.85 m) and solving the equation, we can find the angle θ that the pendulum makes with the vertical when the speed of the bob is 1.4 m/s.