# A sphere of mass 3.0X10^-4 kg is suspended from a cord. A steady horizontal breeze pushes the sphere so that the cord makes a constant angle of 37 degrees with the vertical. find a) the magnitude of that push and b) the tension of the cord.

I looked through my textbook and couldn't find a formula that would fit for part a and part b. What would be the equations I should use?

The wind "push" must equal the cord tension's horizontal component, T sin 37

The weight (M g = 2.94*10^-3 N)must equal the cord tension's vertical component, T cos 37.

It is easiest to solve for T first
T = 2.94*10^-3/cos37 = 3.68*10^-3 N

Then solve for the horizontal "push".

## To solve for the magnitude of the push, you can use the equation T sin(37), where T represents the tension in the cord.

Let's calculate it:

T = 3.68 x 10^-3 N (from the previous step)

Magnitude of push = T sin(37)
Magnitude of push = 3.68 x 10^-3 N x sin(37)
Magnitude of push ≈ 2.23 x 10^-3 N

So, the magnitude of the push is approximately 2.23 x 10^-3 N.

Now, let's move on to part b and solve for the tension in the cord.

The weight of the sphere is equal to the vertical component of the tension, T cos(37).

Let's calculate it:

Weight (Mg) = 2.94 x 10^-3 N

T cos(37) = 2.94 x 10^-3 N

T = (2.94 x 10^-3 N) / cos(37)
T ≈ 3.68 x 10^-3 N

Therefore, the tension in the cord is approximately 3.68 x 10^-3 N.

## To find the magnitude of the horizontal push, we use the formula T sin θ, where T is the tension in the cord and θ is the angle the cord makes with the vertical. In this case, T = 3.68 * 10^-3 N (which we calculated earlier) and θ = 37 degrees.

So, the magnitude of the push (P) is given by P = T sin θ. Plugging in the values, we have P = (3.68 * 10^-3 N) * sin(37 degrees).

To solve this, we can use a calculator to find sin(37 degrees) ≈ 0.6018.

Now we can calculate P = (3.68 * 10^-3 N) * 0.6018. This gives us P ≈ 2.213 * 10^-3 N.

Therefore, the magnitude of the push is approximately 2.213 * 10^-3 N.

Next, we need to find the tension in the cord. The tension (T) can be calculated using the formula T = M g / cos θ, where M is the mass of the sphere and g is the acceleration due to gravity.

In this case, the mass of the sphere (M) is given as 3.0 * 10^-4 kg, and the angle θ is 37 degrees.

The acceleration due to gravity (g) is approximately 9.8 m/s^2.

Now we can substitute these values into the formula to find the tension (T).
T = (3.0 * 10^-4 kg * 9.8 m/s^2) / cos(37 degrees).

Using a calculator to find cos(37 degrees) ≈ 0.7986, we can now calculate T = (3.0 * 10^-4 kg * 9.8 m/s^2) / 0.7986.

This gives us T ≈ 3.68 * 10^-3 N.

Therefore, the tension in the cord is approximately 3.68 * 10^-3 N.

To summarize:
a) The magnitude of the push is approximately 2.213 * 10^-3 N.
b) The tension in the cord is approximately 3.68 * 10^-3 N.