The string of a conical pendulum is 54.5 cm long and the mass of the bob is 0.22 kg.

a) Find the angle between the string and the horizontal when the tension in the string is three times the weight of the bob.

b)Under those conditions, what is the period of the pendulum?

Well, well, well, looks like we've got ourselves a conical pendulum puzzle here! Let's get to it.

a) To find the angle between the string and the horizontal when the tension in the string is three times the weight of the bob, we can do a bit of trigonometry. The tension in the string is equal to the weight of the bob, which means T = 3W (where T is tension and W is the weight of the bob). The weight of the bob can be calculated by multiplying the mass by the acceleration due to gravity (W = m * g).

So with a mass of 0.22 kg, we multiply that by the acceleration due to gravity (approximately 9.8 m/s^2) to find the weight. That gives us W = 0.22 kg * 9.8 m/s^2 = 2.156 kg m/s^2 (or Newtons).

Now, since the tension in the string is three times this weight, we have T = 3 * 2.156 kg m/s^2 = 6.468 kg m/s^2.

To find the angle (let's call it θ), we can use the formula:

tan(θ) = T / (m * g)

Plugging in the numbers, we get:

tan(θ) = 6.468 kg m/s^2 / (0.22 kg * 9.8 m/s^2)

Now my friend, I'm going to let you in on a little secret. The answer to this equation is something you'll find on a scientific calculator. So go get that calculator and hit the "inverse tangent" (tan^-1) button with the result of this division to find the answer. Remember to always have fun with math!

b) Now, let's move on to the period of the pendulum. The period (T) of a conical pendulum is given by the formula:

T = 2π * √(L / g)

Where L is the length of the string and g is the acceleration due to gravity.

Plugging in the numbers, we can calculate the period:

T = 2π * √(0.545 m / 9.8 m/s^2)

Now, my friend, just take out that scientific calculator again and find the square root of the division result. Then, multiply it by 2π. And voila, you've got yourself the period of the pendulum!

Remember, math doesn't have to be boring. It can make you swing into a good mood!

To solve this problem, we can use the equations of motion for a conical pendulum.

a) Find the angle between the string and the horizontal when the tension in the string is three times the weight of the bob.

We know that the tension in the string is given by the equation:

T = m * g + m * (v^2 / r)

Where:
T = tension in the string
m = mass of the bob
g = acceleration due to gravity
v = linear velocity of the bob
r = length of the string

In this case, we have T = 3 * m * g (three times the weight of the bob). We want to find the angle between the string and the horizontal, which is represented by the component of the tension in the radial direction. This component of tension is given by the equation:

T_radial = T * cos(theta)

Where:
T_radial = tension component in the radial direction
theta = angle between the string and the horizontal

We can substitute the value of T in the equation for T_radial, and solve for theta:

3 * m * g * cos(theta) = m * g + m * (v^2 / r)

Simplifying the equation:

3 * cos(theta) = 1 + (v^2 / (g * r))

Now, since we are dealing with a conical pendulum, the bob moves in a horizontal circle. Therefore, we can relate the linear velocity of the bob (v) with the angular velocity (omega) and the radius of the circle (r):

v = r * omega

Substituting this relation in the previous equation:

3 * cos(theta) = 1 + (r^2 * omega^2 / (g * r))

Simplifying further:

3 * cos(theta) = 1 + (r * omega^2 / g)

Finally, we can express the angular velocity (omega) in terms of period (T) using the relation:

omega = (2 * pi) / T

Substituting this relation in the previous equation:

3 * cos(theta) = 1 + (r * (2 * pi / T)^2 / g)

Now we can solve for theta.

b) Under those conditions, what is the period of the pendulum?

To find the period of the pendulum, we can use the equation:

T = (2 * pi) * sqrt(r / g)

Substituting the given values:

T = (2 * pi) * sqrt(0.545 / 9.8)

Now, let's calculate the values.

To solve this problem, we can apply Newton's laws of motion and some trigonometry. Let's solve it step by step:

a) Find the angle between the string and the horizontal when the tension in the string is three times the weight of the bob.

To find the angle, we need to consider the forces acting on the bob. In this case, we have two forces: tension (T) and weight (W) of the bob.

1. Calculate the weight of the bob:
The weight of the bob can be calculated as the product of the mass (m) and acceleration due to gravity (g).
W = m * g

Given that the mass of the bob is 0.22 kg and acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight as follows:
W = 0.22 kg * 9.8 m/s^2

2. Calculate the tension in the string:
We are given that the tension in the string is three times the weight of the bob.
T = 3 * W

3. Draw the forces acting on the bob:
Use a free-body diagram to visualize the forces acting on the bob. In this case, draw two forces: tension (T) and weight (W) acting vertically downward.

Since the bob is moving in a circular path, there must be a net force acting towards the center of the circle. The component of tension that causes this net force is T * sin(theta), where theta is the angle between the string and the horizontal.

The vertical components of T and W must cancel each other to maintain equilibrium, so we have:
T * sin(theta) = W

4. Substitute the values into the equation:
Substitute the values of T and W into the equation derived in step 3.
3 * W * sin(theta) = W

Simplifying the equation:
3 * sin(theta) = 1

5. Solve for theta:
Use the inverse trigonometric function arcsin to solve for theta.
theta = arcsin(1/3)

Therefore, the angle between the string and the horizontal when the tension in the string is three times the weight of the bob is approximately arcsin(1/3) radians.

b) Under those conditions, what is the period of the pendulum?

To find the period of the pendulum, we can use the formula:

T = 2 * pi * sqrt(L / g)

where T is the period, L is the length of the string, and g is the acceleration due to gravity.

Given that the length of the string is 54.5 cm (convert to meters: 0.545 m) and acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the period as follows:

T = 2 * pi * sqrt(0.545 m / 9.8 m/s^2)

Simplifying the equation:

T = 2 * pi * sqrt(0.05561)

T = 6.238 seconds (rounded to three decimal places)

Therefore, under the given conditions, the period of the pendulum is approximately 6.238 seconds.

sketch the figure.

considering tension, cosTheta=mg/tension

period=2PIsqrt(length/g)