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We have inelastic impacts between a ball and a person. The ball is moving and the human is in rest. It hits him first in his Center of mass. In this part question it is imagined that the human stands on ice and he (gets pushed without friction .. yea i know but as i said idealize!) such that we can imply the conversation of momentum.

the part which confused me is this

What happens when the ball hits ABOVE the Center of mass?

So i thought well, maybe the ball will cause a Torque on the horizontal axis of rotation of this human since it is changing its Momentum and we know that the change of momentum / time = force and the distance $d$ where teh ball hits from the Center of mass will give us our Torque $d * F$

Mad
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3 Answers3

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What is the definition of momentum?

Momentum is mass times the velocity of the center of mass.

So Newton's second law says a force or impulse will change the momentum regardless of where this force/impulse is applied.

$$ \begin{aligned} p & = m \,v_{\rm COM} & & & \text{(momentum) }p \\ J & = \Delta p = m \, \Delta v_{\rm COM} & & & \text{(impulse) }J\\ F & = \frac{\rm d}{{\rm d}t} p = m \frac{\rm d}{{\rm d}t} v_{\rm COM} & & & \text{(force) }F \end{aligned} $$

That is all you need to know to describe the motion of the center of mass.

Now for the motion about the center of mass, you need to consider the torque applied about the center of mass. Here is the concept of angular momentum and mass moment of inertia is introduced.

$$ \begin{aligned} L_{\rm COM} & = I_{\rm COM} \,\omega & & & \text{(angular momentum) }L \\ r \times J & = \Delta L_{\rm COM} = I_{\rm COM} \, \Delta \omega & & & \text{(impulse position) }r\\ \tau_{\rm COM} & = \frac{\rm d}{{\rm d}t} L_{\rm COM} & & & \text{(torque) }\tau \end{aligned} $$

If you know the mass moment of inertia $I_{\rm COM}$ is the human about the center of mass, you can calculate the effect of an offset impulse $J$ to the change in rotational speed $\Delta \omega$ with the second equation above.

John Alexiou
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  • So answering the question according to you is as follows:

    1. The ball will change the linear momentum of the human nevertheless since it is applying a force. And follows from the Newtonian law that the object will change movement.

    2. The ball shall cause rotational torque around the Center of mass causing the human to rotate (Either fully or partly)

    Right?

     – Mad May 02 '20 at 06:09
  • @MadSpaceMemer very correct. I think you got it. – John Alexiou May 02 '20 at 16:42
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As you idealized that the person is freely standing on a friction less ice.

The person is free to rotate about center of mass. Therefore when the ball hits the person above center of mass, the person will experience a torque or be rotated if rigid enough about his/her center of mass under the influence of torque caused by the impact force & its distance from center of mass.

Harish Chandra Rajpoot
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If there was no friction at all between the man and the floor say he is standing on very smooth rollers or ice skates, the man would definitely rotate and fall. However, the man may not complete a full rotation or more for the reason that he is reasonably massive and the ball may not carry enough momentum. So in the real world, it will seem as if the man just slid back and fell. But let's say it's a giant boulder with a tremendous amount of force that hit the man. This time he is going to rotate fully, probably more than once before he hits the floor. The reason is that force applied at any (unstable) position other than the COM will definitely produce a torque and the object would rotate about it's own center of mass. Better said, a force produces linear motion if it is applied at the COM but causes rotation when it acts through every other point on the rigid body.

EMMANUEL CHIDERA
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