# Solving Mechanics Problem: Forces, Eqns of Motion, Reactions

• henryc09
In summary, the mass flies off the surface of the hemisphere when its vertical height has decreased by a/3. The only force that the sphere can exert on the mass is the normal force and so when N = 0 the mass flies off.
henryc09

## Homework Statement

A mass m is placed on top of a smooth hemisphere of radius a such that $$\vartheta$$=$$\pi$$/2 (so it is basically on the top of the semicircle, with $$\vartheta$$ being the angle between it and the horizontal).

It is given a very small impulse and as a result begins to slide down one side of the hemisphere under the influence of the gravitational acceleration g.

State the forces acting on the mass, giving their directions, and write down its radial and angular equations of motion in polar coordinates as long as it remains sliding on the sphere.

Find the reaction force between the mass and the surface of the hemisphere as a function of the angle $$\vartheta$$, and hence show the mass flies off the surface of the hemisphere when its vertical height has decreased by a/3.

## Homework Equations

I guess that
a= -r$$\omega$$^2 r^ + r $$\delta$$$$\omega$$/$$\delta$$t $$\vartheta$$^

## The Attempt at a Solution

Only just started this section of the course and so struggling to get my head around a lot of the material. The forces acting are gravity and the normal force, and so I suppose the equation of motion would be:

ma= -mgsin$$\vartheta$$ + N r^ - mgcos$$\vartheta$$ $$\vartheta$$^

Not sure how to express the normal force, but would I be right in saying it flies off where -mgsin$$\vartheta$$ + N < -mr$$\omega$$^2

also when it's at height a/3 sin$$\vartheta$$=1/3

But yeah basically I'm just pretty confused with this topic so far so any help would be appreciated.

You are almost there for the acceleration. Think of the sphere as an incline with continuously changing angle theta. Perpendicular to the "incline" is the radial direction and parallel to the incline is the "theta" direction. What are the components of the weight along these directions?

The mass flies off when the sphere can no longer exert a force on the mass in which case the mass is in free fall. The only force that the sphere can exert on the mass is N. So what do you think the value of N must be when the mass flies off?

right so when the normal force is 0 it flies off.

the component of gravity acting towards the centre is -mgsin$$\vartheta$$ I think.

So the overall centripetal force which is -mr$$\omega$$^2 which equals -mgsin$$\vartheta\vartheta$$ + N and so

N = -mr$$\omega$$^2 + mgsin$$\vartheta$$ and so when N = 0

r$$\omega$$^2 = gsin$$\vartheta$$

not sure where to go now. I guess working out $$\omega$$ as a function of $$\vartheta$$? Although not sure how I'd do that exactly.

Use energy conservation and v = ωR.

ah I see, got it now! Thanks :D

hw excatly did u do it??

## 1. What are the key principles of mechanics that are used to solve problems involving forces and equations of motion?

The key principles of mechanics that are used to solve problems involving forces and equations of motion are Newton's Laws of Motion, the concept of inertia, and the principle of equilibrium. These principles provide a framework for understanding how forces and motion interact and can be applied to various situations to determine the resulting forces and motions.

## 2. How do you solve a mechanics problem involving forces and equations of motion?

To solve a mechanics problem involving forces and equations of motion, you first need to identify all the forces acting on the object and their directions. Then, you can use Newton's Second Law (F=ma) to calculate the net force acting on the object. From there, you can use equations of motion (such as the kinematic equations) to determine the object's position, velocity, and acceleration at different points in time.

## 3. What are reaction forces and how are they related to Newton's Third Law?

Reaction forces are equal and opposite forces that occur in response to an applied force. They are related to Newton's Third Law, which states that for every action, there is an equal and opposite reaction. This means that whenever an object exerts a force on another object, the second object will exert an equal and opposite force back.

## 4. How do you determine the direction and magnitude of a reaction force?

The direction of a reaction force can be determined by considering the direction of the applied force and applying Newton's Third Law. The magnitude of the reaction force is equal to the magnitude of the applied force.

## 5. How can you check the accuracy of your solution to a mechanics problem?

You can check the accuracy of your solution to a mechanics problem by using conservation of energy and/or momentum. If your solution satisfies these principles, then it is likely correct. You can also check your solution using real-world observations or by comparing it to similar problems with known solutions.

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