How Does an Impulse Affect the Motion of a Cube?

[sabatier]

I'd be grateful if someone could help me with the problem below:

A free-floating uniform cube with sides of length 2 is at rest,
aligned with the coordinate axes, when it is subjected to an
impulse along one edge of 3m, where m is its mass. What
is the effect on its motion? (Its inertia tensor is 2/3mI)

Do I calculate the angular velocity or what? And how?

Thanks for any help.

 

[tiny-tim]

Welcome to PF!

Hi sabatier! Welcome to PF!

(how can impulse have dimensions of mass? )

You have to calculate the velocity of the centre of mass, and the angular velocity (including specfiying the axis of rotation) relative to the centre of mass.

 

[astrokreng]

I would approach this problem by first understanding the concept of impulse and its effect on an object. An impulse is a sudden change in momentum, which is the product of an object's mass and velocity. In this case, the cube is subjected to an impulse along one edge, which means there will be a sudden change in its momentum along that direction.

To calculate the effect of this impulse on the motion of the cube, we need to consider both its linear and angular motion. The linear motion can be calculated using the formula: impulse = change in momentum = mass x change in velocity. Since the cube is at rest initially, the change in velocity will be the final velocity after the impulse is applied. So, we can rewrite the formula as: impulse = mass x final velocity.

In this case, the impulse is given as 3m, where m is the mass of the cube. So, we can write the equation as: 3m = m x final velocity. This means that the final velocity of the cube along the direction of the impulse will be 3m/s.

Now, to calculate the angular motion, we need to consider the cube's moment of inertia, which is given as 2/3mI. This represents the object's resistance to changes in its rotational motion. To calculate the angular velocity, we can use the formula: impulse = change in angular momentum = moment of inertia x change in angular velocity. Since we know the impulse and the moment of inertia, we can calculate the change in angular velocity as: 3m = (2/3mI) x change in angular velocity. Solving for change in angular velocity, we get: change in angular velocity = 4.5 rad/s.

Therefore, the effect of the impulse on the motion of the cube will be a linear velocity of 3m/s along the direction of the impulse and an angular velocity of 4.5 rad/s. This means that the cube will start to move in a straight line along the direction of the impulse and will also start to rotate with an angular velocity of 4.5 rad/s.

I hope this explanation helps. It is important to understand the concepts of impulse, momentum, and moment of inertia to solve problems like these. If you need further assistance, please don't hesitate to ask for clarification. Good luck!