A bug starts walking away from the top of a spherical ball of ice and will start to slip when the normal force exerted on it drops to 1/2 of its weight. If the ice ball has a diameter of 35 cm, how far can the bug walk before it begins to slip?
the normal force is ... m * g * cos(Θ)
where Θ is the angle between
... the tangent to the sphere (ball), and the horizontal
the central angle at this point is ... 90º - Θ
... you can draw the diagram to see this
cos(Θ) = 1/2 ... Θ = 60º
so the bug can walk 2/3 of the way down the sphere (60º / 90º)
walking distance ... 2/3 * π * 35 cm / 4
Well, if the bug doesn't want to slip and slide, it better be careful with that normal force! So let's break it down.
First, we need to find the weight of the bug. To do that, we can use the formula: Weight = mass * acceleration due to gravity. But since the question doesn't tell us the mass of the bug, I guess we'll just have to assume it weighs very little next to the ice ball. So, let's just forget about that part.
Now, the normal force exerted on the bug is equal to its weight when it's not slipping. When the normal force drops to 1/2 of its weight, we know it's starting to slip. So, if we let N be the normal force and W be the weight of the bug, we can say that N = W/2.
Next, let's think about the forces acting on the bug. When it's not slipping, it experiences two forces: the normal force and the gravitational force. These two forces, N and W, are equal. So, we have the equation N = W = W/2.
Now back to the ice ball! We know the bug will begin to slip when the normal force drops to 1/2 of its weight. So, let's find the normal force when the bug is about to slip. We know that N = W/2, so we just need to find W/2.
Since we don't know the weight of the bug, we really can't calculate the distance it can walk before it begins to slip. However, I must admit, I'm kind of relieved because bugs slipping on ice balls sounds like a hilarious show to watch!
To determine how far the bug can walk before it begins to slip, we need to find the point on the sphere where the normal force drops to 1/2 of the bug's weight. Let's go through the steps to solve this problem.
Step 1: Find the weight of the bug.
The weight of an object is given by the formula W = m * g, where m is the mass and g is the acceleration due to gravity. Since the mass of the bug is not given, we can assume it to be m kg. The acceleration due to gravity is approximately 9.8 m/s^2. Therefore, the weight of the bug is W = m * 9.8 N.
Step 2: Find the normal force exerted on the bug.
The normal force exerted on the bug is equal to its weight when it is not slipping. So, the normal force is N = W N.
Step 3: Determine when the normal force drops to 1/2 of the bug's weight.
According to the problem, the normal force drops to 1/2 of the weight when N = 1/2W. Substituting the value of W, we have:
N = m * 9.8 N = 1/2(m * 9.8 N)
Step 4: Find the radius of the sphere.
The diameter of the sphere is given as 35 cm. The radius (r) of the sphere is half of the diameter. Therefore, r = 35 cm / 2 = 17.5 cm = 0.175 m.
Step 5: Find the distance the bug can walk before slipping.
The distance the bug can walk before slipping is equal to the circumference of the circle formed by the equator of the sphere at the point where the normal force drops to 1/2 of the bug's weight. The circumference (C) of a circle is given by the formula C = 2πr.
Substituting the value of r, we have:
C = 2π * 0.175 m
Finally, we can calculate the circumference to find the distance the bug can walk before slipping.
To solve this question, we'll use the concept of friction and equilibrium.
When the bug begins to slip, the frictional force between the bug and the ice ball will reach its maximum value. At this point, the normal force exerted on the bug will be equal to half of its weight in order to sustain equilibrium.
Now let's break down the problem into steps:
Step 1: Determine the weight of the bug.
The weight of an object can be calculated using the equation:
Weight = mass × acceleration due to gravity.
Since we don't have the mass of the bug, we can use the fact that weight and mass are directly proportional.
Step 2: Calculate the normal force exerted on the bug when it starts to slip.
The normal force is defined as the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force will be equal to half the weight of the bug.
Step 3: Find the maximum frictional force.
The maximum frictional force is given by the equation:
Frictional force = coefficient of friction × normal force.
The coefficient of friction represents the roughness between the surfaces in contact. Since it is not given in the problem, we'll assume it to be the maximum possible value, which is 1.
Step 4: Determine the force exerted by friction.
The force exerted by friction is equal in magnitude and opposite in direction to the force applied by the bug on the ice ball. Therefore, it will be equal to the maximum frictional force.
Step 5: Calculate the distance the bug can walk before slipping.
The work done by friction is equal to the force exerted by friction multiplied by the distance traveled. In this case, this work will be equal to the change in potential energy of the bug, which is given by the equation:
Change in potential energy = weight × distance.
Solve this equation for distance to find the answer.
Now, let's perform the calculations:
Step 1: Weight of the bug.
As we don't have the mass of the bug, we'll cancel it out by dividing the weight equation by the acceleration due to gravity:
Weight = mass × acceleration due to gravity
Weight/mass = acceleration due to gravity
Let's take the acceleration due to gravity as 9.8 m/s².
Weight/mass = 9.8 m/s²
Step 2: Normal force exerted on the bug.
Since the normal force should be half of the weight, we have:
Normal force = Weight/2
Step 3: Maximum frictional force.
Frictional force = coefficient of friction × normal force
Since the coefficient of friction is assumed to be 1, we have:
Frictional force = 1 × Normal force
Step 4: Force exerted by friction.
The force exerted by friction will be equal to the maximum frictional force calculated in the previous step.
Step 5: Distance the bug can walk before slipping.
The work done by friction is given by the equation:
Work = force exerted by friction × distance
Work = change in potential energy
Therefore,
Force exerted by friction × distance = Weight × Distance
Distance = Weight/force exerted by friction
Now, plug in the values and solve the equation to find the distance the bug can walk before slipping.