A sphere of mass 2.7 x 10^-4 kg is suspended from a cord. A steady horizontal breeze pushes the sphere so that the cord makes a constant angle of 41 degrees with the vertical. Find the magnitude of the push provided by the breeze and Find the tension in the cord.
SEE ABOVE
To find the magnitude of the push provided by the breeze, we need to break down the forces acting on the sphere. The two major forces are the force of gravity and the tension in the cord. The push provided by the breeze can be categorized as a horizontal force.
Let's denote the angle between the cord and the vertical as θ (41 degrees in this case). Since the cord makes an angle with the vertical, the vertical component of the tension in the cord balances the weight of the sphere (force of gravity).
Given the mass of the sphere (m = 2.7 x 10^-4 kg) and the acceleration due to gravity (g ≈ 9.8 m/s^2), we can calculate the weight of the sphere:
Weight = mass × acceleration due to gravity
= m × g
= 2.7 x 10^-4 kg × 9.8 m/s^2
To balance the weight of the sphere, the vertical component of the tension in the cord should be equal to the weight. Therefore, the vertical component of the tension is:
Vertical component of tension = Weight
= 2.7 x 10^-4 kg × 9.8 m/s^2
Now, we can find the horizontal component of the tension (which is the push provided by the breeze). The horizontal component of the tension can be calculated using trigonometric functions. The horizontal component is given by:
Horizontal component of tension = Tension × cos(θ)
So, to find the magnitude of the push provided by the breeze, we need to solve for the horizontal component of the tension. Let's plug in the values:
Magnitude of the push = Horizontal component of tension
= (Tension × cos(θ))
To find the tension in the cord, we can calculate the magnitude of the tension using the Pythagorean theorem. The magnitude of the tension is given by:
Magnitude of the tension = sqrt((Vertical component of tension)^2 + (Horizontal component of tension)^2)
Let's use the calculated values to find the magnitude of the push provided by the breeze and the tension in the cord.