Physics Lab: Measuring the Force on a Bouncing Ball
I like toys — but everyone likes toys, right? But what about physics-toys? Those are the best. Vernier (the science education equipment people) sent me something new to play with — a force plate sensor with a lateral force attachment. The force plate is basically like a bathroom scale that records the force value as a function of time. With the added lateral force sensor, it also measures the force parallel to the floor.
Now it’s time for a physics experiment. I’m going to toss a weighted ball onto the plate and measure both the normal and frictional force on the ball during impact.
Video Analysis
Let’s start off with a look a the motion of the ball as it collides and rebounds. Of course it doesn’t bounce up to its original height, but it should also slow down in the horizontal direction. We should be able to see this with video analysis. Here is the motion the ball during the bounce.
Since I’m going to be using the Vernier force plate, I figured I should also use the Vernier Video Analysis app. The idea is to mark the location of the ball in each frame of the video. If you know the distance scale for the objects in the scene, you can get x, y, and t data for the motion. Notice in the top image that there’s a meter stick on the floor. Now you know why.
I can’t directly get the impact force from the video, but I can estimate the change in momentum. Oh, the mass of the ball is about 2.7 kilograms (we will need that later). Here is a plot of just the vertical position of the ball as a function of time.
There are two quadratic functions here since both before and after the impact the vertical acceleration is -9.8 m/s². I have the coefficients, a b and c for each function. Perhaps it would be useful to write this as:
Looking at the fits above you can see that the acceleration going DOWN is slightly different than the acceleration after the bounce. I think this is fine. What I really want is the vertical velocity right before and right after the impact at t = 0.28 seconds and at t = 0.31 seconds. If I take the derivative of these two position functions I will get the velocity as a function of time.
Plugging in my times, I get the vertical velocities of -4.46 m/s and 2.975 m/s for before and after the impact. Now let’s look at the horizontal motion.
Here the horizontal velocities before and after the collision so that we can just use the slope of these two linear functions. This gives the x-velocities of 2.101 m/s and then 0.9313 m/s.
Putting this all together with the mass of 2.7 kg, I get the following changes in momentum (split into x and y components).
So, how do you change the momentum of an object? Yes, you need a net force. The momentum principle states:
We have the change in momentum, so let’s switch over the to the force and time measurements.
Force Plate
I’m using the Go Direct Force Plate along with the Lateral Force attachment. This means that I can collect force as a function of time. Since the impact happens quite quickly, I needed to set the data rate to a large value. After playing around with the experiment, it seems like 300 samples per second worked well enough. Here’s the both the upward (y) force and the horizontal (x) force.
We can write the momentum principle as:
If you know the force over some time interval, you can calculate the change in momentum. Of course in our case the force is not constant. It’s fine. We can break the impact into many smaller time intervals (1/300 seconds). With a super short time interval we can make the approximation that the force is constant and then calculate a super tiny change in momentum. Doing this for all the time intervals means that we can just add up all the changes to get the total change in momentum. This is actually a numerical integration — the data collection software (Vernier’s Graphical Analysis) can do this for us.
The graph above already has the results of the numerical integral. Check it out. The integral for the vertical force is 21.786 N*s. Oh, look — the change in vertical momentum was 20.08 N*s. That’s actually pretty close. For the horizontal motion the integral gives -3.5 N*s compared to the change in x-momentum of -3.16 N*s.
The numbers don’t match perfectly, but I’m going to count this as a win.
Comments (and HW)
- Why does the horizontal velocity decrease after the impact? It’s because there is a frictional force on the ball pushing in the opposite direction of the motion of the ball.
- I could probably get better velocity data by using a video frame rate of 60 fps (or even higher). Of course with higher frame rates you usually need more light on the object (in case you want to try that).
- We have values for the vertical force (the normal force) and the horizontal force (frictional force). Can you use this to get a minimum value for the coefficient of friction? Hint: yes you can.
- I think you could repeat this experiment with the force plate at an angle instead of flat. Then you could just drop the ball onto the angled plate and see it bounce off to the side. This might be a little easier since you don’t have to aim the ball right at the sensor when it’s thrown. Maybe I’m just a bad ball thrower.
- How much kinetic energy was lost during the collision?
- Here’s a cool one — drop a spinning ball onto the plate. Record a video and look at the change in rotational motion of the ball along with the horizontal impulse. That would be cool.
