a rice grain is sitting on top of an apple, it slides down. where does it come off the apple? assuming the apple is a frictionless sphere. Find the angle with respect to the vertical diameter of the apple.

A person desires to reach a point that is 5.20 km from her present location and in a direction that is 43° north of east. However, she must travel along streets that are oriented either north-south or east-west. What is the minimum distance she could travel to reach her destination?

To determine where the rice grain comes off the apple and the angle it makes with respect to the vertical diameter, we can use the concept of the critical angle.

When the rice grain is on the top of the apple, gravity acts vertically downwards, and there is no friction or external force. Therefore, the rice grain only experiences a downward force.

As the rice grain starts sliding down, it undergoes an inclined motion due to the gravitational force. At some point, it will reach a critical angle where the gravitational force pulling it downwards becomes equal to the centrifugal force pulling it outwards.

To find this critical angle, we can equate the gravitational force with the centrifugal force. The gravitational force can be broken down into two components: one parallel to the inclined surface and one perpendicular to it.

The parallel component of the gravitational force is given by: F_parallel = m * g * sin(theta), where m is the mass of the rice grain, g is the acceleration due to gravity, and theta is the angle with respect to the vertical diameter.

The centrifugal force acting outwards is given by: F_centrifugal = m * omega^2 * r, where omega is the angular velocity of the apple (since it's assumed to be spinning) and r is the distance of the grain from the center of the apple.

Setting F_parallel equal to F_centrifugal, we have:

m * g * sin(theta) = m * omega^2 * r

Canceling out the mass (m) and rearranging the equation, we get:

sin(theta) = (omega^2 * r) / g

To find the angle, we can take the inverse sine (sin⁻¹) of both sides:

theta = sin⁻¹((omega^2 * r) / g)

Note that the value of omega would depend on the spinning rate of the apple, and r would depend on the distance of the grain from the center of the apple.

Therefore, to calculate the exact angle, you need to know the values of omega and r. Without this information, it is not possible to determine the specific angle.