Impulse is defined as the integral of a force acting on an object, with respect to time. This means that impulse contains the product of force and time. Impulse changes the momentum of an object. As a result, a large force applied for a short period of time can produce the same momentum change as a small force applied for a long period of time. An impulse can act on an object to change either its linear momentum, angular momentum, or both.

In many real life problems involving impulse and momentum, the impulse acting on a body consists of a large force acting for a very short period of time – for example, a hammer strike, or a collision between two bodies.

The following problem illustrates the principle of impulse and momentum.

A solid ball of mass

Assume that the ball pivots about the tip of the bump during, and after impact.

Set up a schematic of this problem, as shown, along with sign convention. Assume the center of mass

Let

We can treat this as a planar motion problem. It can be solved using the principle of impulse and momentum. Since this problem combines translation and rotation we must apply the equations for linear momentum and angular momentum.

In an impact of very short time duration (say, between an initial time

is dominated by the impact force

For planar motion in the

Where:

The integrals are the impulse terms.

Since the ball initially rolls without slipping,

The negative sign accounts for the fact that positive angular velocity means the ball rolls to the left (in the negative

Since the ball initially rolls on a flat horizontal surface,

Immediately after impact the ball pivots about point

Since the ball pivots about point

From geometry we can write

and

Substituting equations (3)-(6) into equations (1) and (2) we get

Now, for planar motion, the equation for impulse and angular momentum is:

Where:

Σ

Now, referring to the figure above we can write

Thus,

This can be written as

Substituting the above equation into equation (9) we get

We can combine the three equations (7), (8), and (10) to solve for the three unknowns:

However, we only need to know

Now, by geometry

The moment of inertia of the solid ball about

Substitute the above two equations into the equation for

This is the angular velocity of the ball immediately after impact.

We now need to find the necessary initial angular velocity

We can use conservation of energy after impact since the only force that does work on the ball after impact is the gravitational force (which is a conservative force).

In the figure below, let’s define position 1 of the ball as its position immediately after impact. Due to the very short time of impact, this position very closely coincides with the position of the ball just as it touches the tip of the bump at point

In the figure below, let’s define position 2 of the ball as its top-most position while it pivots about point

Between positions 1 and 2, the equation for conservation of energy is:

Where:

The initial kinetic energy of the ball is:

The initial potential energy of gravity acting on the ball (measured from the datum) is:

The final kinetic energy of the ball is:

The final potential energy of gravity acting on the ball (measured from the datum) is:

Substituting equations (13)-(16) into equation (12) we get

From before,

and

Substitute the above two equations into equation (17), and we get

Solving for

Substituting this equation into equation (11) and solving for

This is the minimum initial angular velocity so that the ball just makes it over the bump.

The minimum initial speed of the ball

It’s interesting that when solving for

If we wanted to solve for

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