A ball of mass m = 1.65 kg is released from rest at a height...

A ball of mass $m = 1.65$ kg is released from rest at a height $h = 60.0$ cm above a light vertical spring of force constant $k$ as in Figure [a] shown below. The ball strikes the top of the spring and compresses it a distance $d = 8.90$ cm as in Figure [b] shown below. Neglecting any energy losses during the collision, find the following. (a) Find the speed of the ball just as it touches the spring. m/s (b) Find the force constant of the spring. kN/m

Transcript

0:00 Hello.

00:02 So in this problem, we have this spring here.

00:07 And we have a ball that is placed at a distance, 76 centimeters above the beginning of the spring.

00:16 76 centimeters.

00:19 The ball started from rest and it was released.

00:22 So let's start first by the first question.

00:25 What is the velocity of the ball right before it touched the spring? so the point is here when the ball is higher it has such potential energy and when it starts to move it loses this potential energy and converts it to kinetic energy so simply it means that the potential energy at the beginning is equal to the kinetic energy at the end so the initial potential energy is equal to the final kinetic energy is equal to the final kinetic energy or like in other words, we can say that the change in the potential energy in negative, of course, is equal to that change in kinetic energy for positive.

01:13 So what is lost as potential energy is again as kinetic energy.

01:17 So here, the loss in potential energy, the potential energy is defined as the m, the mass times the gravitational acceleration, times that change in height.

01:28 So let's assume that we have our zero point set here, or in other words, that change is definitely negative 76 centimeters.

01:38 We can also multiply times 10 to the negative 2 to switch to meters instead of centimeters.

01:45 And of course, we have this negative sign here.

01:47 On the other hand, this is equal to the changing kinetic energy, which is half, multiplied by the mass multiplied by the change in the velocity.

01:56 So here the velocity started as zero.

02:00 So the change of the velocity is squared.

02:02 That's the final velocity.

02:03 So that's actually what we're searching for.

02:06 So let's start to plug in numbers.

02:09 So here we would have 1 .5 and this negative cancels with this negative inside, multiplied by 9 .8 multiplied by 0 .76.

02:19 That's equal to half multiplied by 1 .5, the velocity is squared.

02:26 From here, we note that the velocity squared is equal to 11 .172 divided by 0 .5 times 1 .5, which is equal to 44 .688.

02:41 Taking the square root, we note that the velocity is 6 .685 meters per second...

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