A 0.480 kg pendulum bob passes through the lowest part of its path at a speed of 3.10 m/s. (b) When the pendulum reaches its highest point, what angle does the cable make with the vertical?

The cable of the pendulum is 80 cm or .8m.

Well, isn't this a swingin' question! Let's put on our physics party hats and find the answer.

First, let's determine the total energy of the pendulum at the lowest point (let's call it point A). We can use the formula for kinetic energy: KE = (1/2)mv^2, where m is the mass of the bob and v is its velocity. Plugging in the values, we get KE(A) = (1/2)(0.480 kg)(3.10 m/s)^2 = 2.2672 J.

Now, at the highest point (let's call it point B), the pendulum is at its maximum height, so it has only potential energy. Using the formula for gravitational potential energy, PE = mgh, where h is the height above the reference point (in this case, the lowest point of the swing). We can set the total energy at point A equal to the potential energy at point B: KE(A) = PE(B).

Since h is the highest point minus the lowest point, we have PE(B) = (0.480 kg)(9.8 m/s^2)(h - 0.8 m). So, we can set up the equation 2.2672 J = (0.480 kg)(9.8 m/s^2)(h - 0.8 m) and solve for h.

After some dancing with numbers, we find h ≈ 1.253 m.

Now, we're ready to find the angle θ that the cable makes with the vertical at point B. We know that cos(θ) = adjacent/hypotenuse, and in this case, the adjacent side is 0.8 m (the length of the cable) and the hypotenuse is h.

So, cos(θ) = 0.8 m / 1.253 m ≈ 0.638, and solving for θ gives us θ ≈ arccos(0.638) ≈ 50.35 degrees.

So, the angle the cable makes with the vertical at the highest point is approximately 50.35 degrees. See? Physics isn't so intimidating when you add a pinch of humor into the mix!

To find the angle the cable makes with the vertical when the pendulum reaches its highest point, we can use the conservation of mechanical energy.

The mechanical energy of the pendulum is conserved as it swings between the lowest and highest points. This means that the sum of its kinetic energy and potential energy at the lowest point is equal to the sum of its kinetic energy and potential energy at the highest point.

At the lowest point, all of the pendulum's energy is in the form of kinetic energy. So we can calculate the kinetic energy at the lowest point using the formula:

KE = 1/2 * m * v^2

where KE is the kinetic energy, m is the mass of the bob, and v is its velocity.

Given that the mass of the pendulum bob is 0.480 kg and its velocity at the lowest point is 3.10 m/s:

KE_lowest = 1/2 * 0.480 kg * (3.10 m/s)^2
= 1/2 * 0.480 kg * 9.61 m^2/s^2
= 2.3136 J

At the highest point, all of the pendulum's energy is in the form of potential energy, given by:

PE = m * g * h

where PE is the potential energy, m is the mass of the bob, g is the acceleration due to gravity (approximated as 9.8 m/s^2), and h is the height of the bob above the lowest point.

Since the cable length is 0.8 m, the height of the bob above the lowest point is 0.8 m.

PE_highest = 0.480 kg * 9.8 m/s^2 * 0.8 m
= 3.7632 J

Using the conservation of mechanical energy, we can equate the kinetic energy at the lowest point (KE_lowest) and the potential energy at the highest point (PE_highest):

KE_lowest = PE_highest

2.3136 J = 3.7632 J

Now, we can solve for the angle the cable makes with the vertical at the highest point. The potential energy at the highest point is given by:

PE_highest = m * g * h

where h is the height of the bob above the lowest point and the angle θ.

PE_highest = m * g * h * cos(θ)

Since the total potential energy at the highest point is 3.7632 J, we can write:

3.7632 J = 0.480 kg * 9.8 m/s^2 * 0.8 m * cos(θ)

Simplifying the equation:

3.7632 J = 3.7632 cos(θ)

Dividing by 3.7632 on both sides:

1 = cos(θ)

Now, to find θ, we need to take the inverse cosine (cos^-1) of 1.

θ = cos^-1(1)

The inverse cosine of 1 is 0 radians or 0 degrees.

Therefore, the angle the cable makes with the vertical when the pendulum reaches its highest point is 0 radians or 0 degrees.

To find the angle that the cable makes with the vertical when the pendulum reaches its highest point, we can use the law of conservation of mechanical energy.

The law of conservation of mechanical energy states that the total mechanical energy of an object remains constant as long as only conservative forces are acting on it. In this case, the only conservative force acting on the pendulum bob is gravity.

At the lowest part of the pendulum's path, all of its energy is in the form of kinetic energy. The formula for kinetic energy is:

KE = (1/2) * m * v^2

Where KE is the kinetic energy, m is the mass of the pendulum bob, and v is the velocity of the pendulum bob.

At the highest point of the pendulum's path, all of the energy is in the form of potential energy. The formula for potential energy due to gravity is:

PE = m * g * h

Where PE is the potential energy, m is the mass of the pendulum bob, g is the acceleration due to gravity, and h is the height of the pendulum bob above the lowest point.

Since the total mechanical energy is conserved, we can equate the initial kinetic energy to the final potential energy:

KE = PE

(1/2) * m * v^2 = m * g * h

We can cancel out the mass term:

(1/2) * v^2 = g * h

Now, we can solve for the height (h) at the highest point of the pendulum's path:

h = (1/2) * v^2 / g

Substituting the values given in the problem, we have:

h = (1/2) * (3.10 m/s)^2 / 9.8 m/s^2

h = 0.4839 m

Now, we can calculate the angle (θ) that the cable makes with the vertical using trigonometry:

sin(θ) = h / cable length

θ = sin^(-1)(h / cable length)

θ = sin^(-1)(0.4839 m / 0.8 m)

θ ≈ 0.6659 rad

Therefore, the angle that the cable makes with the vertical when the pendulum reaches its highest point is approximately 0.6659 radians.