A flat circular mirror of radius 0.150 m is lying on the floor. Centered directly above the mirror , at a height of 0.750 m, is a small light source. Calculate the diameter of the bright circular spot formed on the 2.90 m high ceiling by the light reflected from the mirror.

To calculate the diameter of the bright circular spot formed on the ceiling, we need to analyze the mirror reflection.

Step 1: Draw a diagram to visualize the situation. Draw a circle to represent the mirror on the floor, and label its radius as 0.150 m. Mark the position of the light source directly above the mirror and label its height as 0.750 m. Draw a line extending from the light source to the mirror, representing the incident ray.

Step 2: Apply the law of reflection. The angle of incidence (θ) is equal to the angle of reflection (θ). In this case, the incident ray is perpendicular to the floor, so the angle of incidence is 0 degrees, resulting in an angle of reflection of 0 degrees as well.

Step 3: Calculate the distance between the mirror and the ceiling. The total distance from the light source to the ceiling is the sum of the height of the light source (0.750 m) and the height of the ceiling (2.90 m). Thus, the distance from the mirror to the ceiling is 0.750 m + 2.90 m = 3.65 m.

Step 4: Determine the diameter of the bright spot on the ceiling. On the diagram, mark the point where the reflected ray intersects the ceiling. Extend a line from the mirror to this point, representing the reflected ray. The diameter of the bright spot is equal to twice the distance between the point of intersection and the mirror.

Step 5: Calculate the distance between the point of intersection and the mirror. We can use similar triangles to determine this distance. Notice that the distance ratio between the point of intersection and the mirror is the same as the distance ratio between the ceiling and the mirror.

Let x be the distance between the point of intersection and the mirror, and 3.65 m be the distance between the ceiling and the mirror. Using the similar triangle property, we can set up the following equation:

x / 0.150 m = 3.65 m / (0.150 m + x)

Step 6: Solve the equation to find the value of x. Cross-multiply and simplify the equation:

x * (0.150 m + x) = 3.65 m * 0.150 m
0.150 m * x + x^2 = 0.5475 m^2
x^2 + 0.150 m * x - 0.5475 m^2 = 0

This is a quadratic equation, which we can solve using the quadratic formula. The solutions will give us the possible values for x. In this case, we only consider the positive root since negative distance doesn't make sense in this context.

x = [-0.150 m + sqrt((0.150 m)^2 - 4 * 1 * (-0.5475 m^2))] / 2 * 1

Simplifying this equation will give us the value of x, which represents the distance between the point of intersection and the mirror.

Step 7: Once you have the value of x, multiply it by 2 to get the diameter of the bright spot on the ceiling.

To calculate the diameter of the bright circular spot formed on the ceiling, we can use the law of reflection. According to the law of reflection, the angle of incidence is equal to the angle of reflection.

Step 1: Determine the position of the light source in relation to the mirror. The light source is centered directly above the mirror at a height of 0.750 m.

Step 2: Calculate the distance between the light source and the mirror's surface. Since the light travels in a straight line, the distance from the light source to the mirror is equal to the height of the light source, which is 0.750 m.

Step 3: Determine the distance between the mirror and the ceiling. This is given as 2.90 m.

Step 4: Calculate the angle of incidence. The angle of incidence is the angle that the light ray makes with the normal line (a line perpendicular to the surface of the mirror) at the point where the light strikes the mirror. In this case, the angle of incidence can be calculated using trigonometry. The tangent of the angle of incidence is equal to the height of the light source divided by the distance between the light source and the mirror's surface.

tan(theta) = height of light source / distance to mirror
tan(theta) = 0.750 m / 0.750 m
tan(theta) = 1

Since the tangent of the angle of incidence is 1, the angle of incidence is equal to 45 degrees.

Step 5: Calculate the angle of reflection. Since the angle of reflection is equal to the angle of incidence, the angle of reflection is also 45 degrees.

Step 6: Calculate the distance between the mirror's surface and the ceiling. This is equal to the height of the ceiling minus the height of the light source.

distance between mirror and ceiling = 2.90 m - 0.750 m
distance between mirror and ceiling = 2.15 m

Step 7: Calculate the diameter of the spot formed on the ceiling. This can be done by considering the geometry of the problem. The distance between the mirror's surface and the ceiling is the base of a right-angled triangle, and the diameter of the spot on the ceiling is the hypotenuse of this triangle. Using trigonometry, the diameter can be calculated as follows:

diameter = 2 * (distance between mirror and ceiling) * tan(angle of reflection)
diameter = 2 * 2.15 m * tan(45 degrees)
diameter ≈ 4.30 m

Therefore, the diameter of the bright circular spot formed on the ceiling by the light reflected from the mirror is approximately 4.30 m.