Matter and Motion

Physics Lab 8

Luki Goldschmidt and Costanzo Allione

November 28, 2000

The Projection of a Body Acted on by Friction

Introduction:

When a body is projected along a smooth horizontal surface with an elastic cord, which behaves according to Hook’s Law, the distance traveled by the body depends on several factors. Our goal in this lab is to quantitatively analyze those factors and find their relationship to the distance traveled.

The free-body diagram on the right shows the forces acting on the projected body. Assuming the object is on a horizontal surface, the gravitational force w and the normal force N balance and will not influence the distance traveled by the body. The energy available from the elastic cord upon release will do the work (F•d) on the body to move it along the surface. Assuming we neglect air resistance and the energy lost due to other factors such as sound, all available energy will be used up by friction f bringing the body to a stop after a distance d. Consequently, the following equation relates all factors to the distance d:

ESPRING = WFRICTION

½ k x2 = N µK d = m g µK d (1)

Where k = spring constant, x = extension of the spring from its original length, m = mass of the projected object, µK = friction coefficient for the given surface and d = distance traveled by the projected body.

As stated in equation (1), increasing the spring constant k or the spring extension x will increase the energy available to project the body and hence the distance traveled d will be greater. When the mass of the projected body m or the friction coefficient µK is increased, more energy is lost due to friction and hence the distance traveled d will be smaller.

We will try to demonstrate that the squared extension of the spring x2 is directly proportional to the distance d traveled by the projected body. Further, the mass of the projected body m is inversely proportional to the traveled distance d.

Experimental:

We have decided to vary m and x respectively while keeping the other variables constant and measure the distance traveled d.

The behavior of a moderately stretched rubber band was investigated to verify that it acts according to Hook’s Law. The required force to stretch the band for each extension was measured using a force transducer.

A round body was projected along a smooth surface by releasing the band as quickly and precisely as possible. The means of releasing the band can greatly influence d if energy is lost due to spin of the body or if friction in the release mechanism is present during the release. The distance traveled by the projected body was measured using a meter stick. The experiment was repeated for varying extensions of the spring x and varying masses of the body m.

Results:

A: Determination of the spring constant k.

Extension (m)

± 0.001

Force (N)

± 0.02

0.015

3.3

0.027

8.0

0.042

16.0

0.054

19.3

0.065

23.0

As expected, the graph of Force vs. Extension is linear. The relationship is described by the equation F = k x making the slope on the graph equal to the constant k.

The spring constant for our experimental setup was determined to be 401 N/m (blue rubber band, 3rd post).

 

 

 

 

 

B: Varying Extension x.

Mass m (g)

Extension (m)

Force F (N)

Average d (m)

Std. Diviation

0.0567 ± 0.001

0.015

3.30

0.189

0.0130

0.0567± 0.001

0.027

7.99

0.617

0.0529

0.0567± 0.001

0.042

16.0

1.353

0.1366

0.0567± 0.001

0.054

19.3

2.028

0.0898

0.0567± 0.001

0.065

23

2.136

0.1683

Raw Data: Measured Distance (m) for each extension x

0.58

1.97

1.43

0.18

2.18

0.58

2.07

1.31

0.19

2.00

0.59

1.97

1.36

0.19

2.16

0.60

2.05

1.44

0.20

2.22

0.61

2.06

1.27

0.20

2.12

0.62

2.05

1.32

0.62

0.64

0.64

0.64

0.65

0.66

 

We have hypothesized that the distance traveled by the projected body will be proportional to the extension squared applied on the spring (band). The graph on the right validates this statement.

The relationship on the graph is:

d = x2 • k / (2 m g µK)

Hence the slope is:

slope = k / (2 m g µK)

Thus, we can calculate µK from these data as follows:

µK = k / (2 m g slope)

The friction coefficient for our surface (CAL floor) was calculated to be

µK = 401 / (2 x 0.0567 x 9.81 x 720) = 0.500.

 

 

 

C: Varying Mass m.

Mass m (kg)

Extension (m)

Force F (N)

Average d (m)

Std. Diviation

0.0567± 0.001

0.054

19.3

2.03

0.0898

0.0687± 0.001

0.054

19.3

1.66

0.0748

0.0821± 0.001

0.054

19.3

1.55

0.1479

0.1154± 0.001

0.054

19.3

0.92

0.1767

0.1412± 0.001

0.065

0.054

0.82

0.0804

Raw Data: Measured Distance (m) for each extension x

1.97

1.61

1.51

0.84

0.80

2.07

1.63

1.62

0.81

0.77

1.97

1.68

1.63

0.89

0.82

2.05

1.69

1.52

1.02

0.84

2.06

1.69

1.46

0.99

0.89

2.05

0.98

0.82

 

Our hypothesis that the distance traveled d is inversely proportional to the mass of the projected body m also appears to be valid.

The relationship on the graph is:

d = 1/m • (kx2 / 2gµK)

Hence the slope is:

slope = kx2 / 2gµK

The friction coefficient from these data was calculated to be:

µK = kx2 / (2 g slope) = 401 x (0.054)2 / (2 x 9.81 x 0.1162) = 0.513

Conclusion:

We have succeeded to validate our hypothesis that the distance traveled by the projected object d in this system is proportional to the extension of the spring squared x2, and inversely proportional to the mass of the projected object m. The graph of distance vs. extension squared (graph 2) is linear with a regression coefficient of 0.995. Graph 3, distance vs. 1/mass also shows good linearity (R2=0.975) with the trendline crossing through all error bars. The spring constant for our system was calculated to be 401 N/m from graph 1.

Further, two data sets (varying extension and varying mass) agree on the value for the calculated friction coefficient µK to be about 0.51 (CAL floor). This value seems high compared to results obtained from other teams, however, the surface varied from team to team.