To find the angle θ in the Figure, we can use the concept of projectile motion.

1. First, let's analyze the motion of the block just before it leaves the surface at point C. At this point, the block has a velocity of 0.5√gr, where g is the acceleration due to gravity and r is the distance from point C to the highest point of the trajectory.

2. Since the surface is frictionless, the only force acting on the block is the force of gravity. This force can be resolved into two components: one parallel to the surface and one perpendicular to the surface.

3. The component of the force of gravity parallel to the surface does not affect the motion of the block, as there is no friction. Therefore, the only force that affects the motion of the block is the perpendicular component of the force of gravity.

4. The perpendicular component of the force of gravity provides the centripetal force required to keep the block moving in a circular path. This force can be calculated using the equation F = m * a, where F is the force, m is the mass of the block, and a is the centripetal acceleration.

5. The centripetal acceleration can be calculated using the equation a = v^2 / r, where v is the velocity of the block and r is the radius of the circular path.

6. Substituting the given values, we have a = (0.5√gr)^2 / r = 0.25gr / r.

7. The centripetal force can be calculated using the equation F = m * a. Since the only force acting on the block is the perpendicular component of the force of gravity, we have F = m * 0.25gr / r.

8. The angle θ can be determined by considering the forces acting on the block at point C. The perpendicular component of the force of gravity is balanced by the normal force exerted by the surface. Therefore, we have F = m * 0.25gr / r = m * g * sin(θ).

9. Simplifying the equation, we have 0.25gr / r = g * sin(θ).

10. Canceling out the mass and the acceleration due to gravity, we have 0.25r = r * sin(θ).

11. Dividing both sides of the equation by r, we have 0.25 = sin(θ).

12. Taking the inverse sine of both sides of the equation, we have θ = sin^(-1)(0.25).

13. Using a calculator, we find that θ ≈ 14.48 degrees.

Therefore, the angle θ in the Figure is approximately 14.48 degrees.

Question

To find the angle θ in the Figure, we can use the concept of projectile motion.

1. First, let's analyze the motion of the block just before it leaves the surface at point C. At this point, the block has a velocity of 0.5√gr, where g is the acceleration due to gravity and r is the distance from point C to the highest point of the trajectory.

2. Since the surface is frictionless, the only force acting on the block is the force of gravity. This force can be resolved into two components: one parallel to the surface and one perpendicular to the surface.

3. The component of the force of gravity parallel to the surface does not affect the motion of the block, as there is no friction. Therefore, the only force that affects the motion of the block is the perpendicular component of the force of gravity.

4. The perpendicular component of the force of gravity provides the centripetal force required to keep the block moving in a circular path. This force can be calculated using the equation F = m * a, where F is the force, m is the mass of the block, and a is the centripetal acceleration.

5. The centripetal acceleration can be calculated using the equation a = v^2 / r, where v is the velocity of the block and r is the radius of the circular path.

6. Substituting the given values, we have a = (0.5√gr)^2 / r = 0.25gr / r.

7. The centripetal force can be calculated using the equation F = m * a. Since the only force acting on the block is the perpendicular component of the force of gravity, we have F = m * 0.25gr / r.

8. The angle θ can be determined by considering the forces acting on the block at point C. The perpendicular component of the force of gravity is balanced by the normal force exerted by the surface. Therefore, we have F = m * 0.25gr / r = m * g * sin(θ).

9. Simplifying the equation, we have 0.25gr / r = g * sin(θ).

10. Canceling out the mass and the acceleration due to gravity, we have 0.25r = r * sin(θ).

11. Dividing both sides of the equation by r, we have 0.25 = sin(θ).

12. Taking the inverse sine of both sides of the equation, we have θ = sin^(-1)(0.25).

13. Using a calculator, we find that θ ≈ 14.48 degrees.

Therefore, the angle θ in the Figure is approximately 14.48 degrees.

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