A sphere of mass 3.3 10-4 kg is suspended from a cord. A steady horizontal breeze pushes the sphere so that the cord makes an angle of 41° with the vertical when at rest
(a) Find the magnitude of that push.
(b) Find the tension in the cord.
To find the magnitude of the push and the tension in the cord, we can apply Newton's laws of motion.
Let's denote the mass of the sphere as m = 3.3 x 10^(-4) kg and the angle the cord makes with the vertical as θ = 41°.
(a) To find the magnitude of the push, we need to calculate the horizontal component of the gravitational force acting on the sphere.
The gravitational force is given by Fg = m * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
The horizontal component of the gravitational force is Fgx = Fg * sin(θ).
So, Fgx = (m * g) * sin(θ).
Substituting the given values, we have Fgx = (3.3 x 10^(-4) kg * 9.8 m/s^2) * sin(41°).
Calculating this value, we find that the magnitude of the push is approximately 1.974 x 10^(-3) N.
(b) The tension in the cord can be found by considering the vertical equilibrium of forces.
The vertical component of the gravitational force is Fgy = Fg * cos(θ).
The tension in the cord, T, can be equated to Fgy:
T = Fgy = Fg * cos(θ).
Substituting the given values, we have T = (3.3 x 10^(-4) kg * 9.8 m/s^2) * cos(41°).
Calculating this value, we find that the tension in the cord is approximately 2.585 x 10^(-4) N.
To solve this problem, we can consider the forces acting on the sphere when it is at rest and find the components of these forces along the horizontal and vertical directions.
(a) Finding the magnitude of the push:
When the sphere is at rest, the gravitational force acting on it is balanced by the tension in the cord. The sphere is in equilibrium, so the sum of the forces in the vertical direction is zero.
Let's assume that the gravitational force on the sphere is given by Fg and the push from the breeze is given by Fp.
Therefore, we can write the equation for forces in the vertical direction as:
Fg - T = 0 (1)
Where T is the tension in the cord.
The gravitational force is given by:
Fg = mg
Substituting the given values, m = 3.3 * 10^(-4) kg and g = 9.8 m/s^2, we can calculate Fg.
Once we have Fg, we can substitute it into equation (1) to solve for T. By doing so, we will get the tension in the cord.
(b) Finding the tension in the cord:
As mentioned earlier, the sum of forces in the vertical direction is zero when the sphere is at rest. This can be represented by equation (1).
Since the sphere is also subjected to a horizontal breeze push, there is a force acting in the horizontal direction. Let's call it Fp.
To find the magnitude of Fp, we need to consider the component of Fp along the vertical direction. We can define the angle made by the cord with the vertical as θ. Since the vertical and the cord form a right triangle, we can apply trigonometric functions to find the component of Fp along the vertical direction.
The vertical component of the push force can be calculated as:
Fp_vertical = Fp * sin(θ)
Substituting the given value of θ, we can find the vertical component of the push force.
Finally, we can substitute the value of Fp_vertical into equation (1), alongside the value of Fg, to solve for the tension in the cord (T).
By following these steps, you should be able to calculate both the magnitude of the push (Fp) and the tension in the cord (T).