Exercises

 

In the following experiments, the experimental data needs to be converted into a linear graph.  In each case, answer the four questions:

 

1.      What do we plot on the horizontal or x axis?

2.      What do we plot on the vertical or y axis?

3.      What should the slope of the straight line be?

4.      What should the y-intercept of the straight line be?

 

Also, if necessary, do uncertainty propagation to find the uncertainty in the experimental quantity you are looking for.

 

a.   The velocity of a glider on an inclined frictionless airtrack is given by the formula    

 

v = v0 + at.

 

The velocity v is measured at various times t.  We are interested in measuring the acceleration.

 

Answer

 

 

So a = m and Da = Dm

 

 

           

 

b.  The magnetic field B at the centre of a circular loop of wire of radius R and carrying current I is given by the formula

 

B = m0I/2pR.

 

where m0 is the permeability of free space. While the current I is kept constant, the radius R of the circular loop is varied and the magnetic field B is measured. We are interested in measuring m0 and comparing it to its accepted value m0 = 1.2566 × 10-6 T-m/A. 

 

Answer

 

 

So m0 = 2pm/I and Dm0/m0 = 2p (Dm/m + DI/I)

 

 

 

c.   The magnetic field B at the centre of a circular loop of wire of radius R and carrying current I is given by the formula

 

B = m0I/2pR.

 

where m0 is the permeability of free space. While the radius R of the circular loop is kept constant, the current I in the loop is varied and the magnetic field B is measured. We are interested in measuring m0 and comparing it to its accepted value m0 = 1.2566 × 10-6 T-m/A. 

 

Answer

 

 

So m0 = 2pmR and Dm0/m0 = 2p (Dm/m + DR/R)

 

 

 

d.  The force of attraction F between two straight wires separated by a distance r and each carrying an equal but opposite current I is given by the formula 

 

F = m0I2L/2pr2.

 

where m0 is the permeability of free space. For fixed length L of wire and current strength I, the force F between the wires is measured as their separation r is changed.  We are interested in measuring m0 and comparing it to its accepted value m0 = 1.2566 × 10-6 T-m/A. 

 

Answer

 

 

So m0 = 2pm/I2L and Dm0/m0 = 2p (Dm/m + 2DI/I + DL/L)

 

 

 

 

e.  The acceleration, a, of a cart on an inclined plane is given by the formula

 

a = gsinq

 

 

 

where g is the acceleration due to gravity and q is the angle that the incline makes to the horizontal.  As the angle q is changed, the acceleration a is measured.  We are interested in measuring g and comparing it to its accepted value g = 9.81 m/s2. 

 

Answer

 

 

So g = m and Dg = Dm

 

 

 

f.    The velocity v of a ball bearing on a banked curve is given by the formula  

 

v2 = Rgtanq

 

where g is the acceleration, R is the radius of curvature of the banked curve, and q is the angle at which the curve banks, i.e. makes to the horizontal.  The velocity v is measured as the banking angle q is changed.  The radius of curvature R is kept constant.  We are interested in measuring g and comparing it to its accepted value g = 9.81 m/s2.

 

Answer

 

 

So g = m/R and Dg/g = (Dm/m + DR/R)

 

 

 

g.  The velocity v of a ball bearing on a banked curve is given by the formula  

 

v2 = Rgtanq

 

where g is the acceleration, R is the radius of curvature of the banked curve, and q is the angle at which the curve banks, i.e. makes to the horizontal.  The velocity v is measured as the radius of curvature R is changed.  The banking angle q is kept constant.  We are interested in measuring g and comparing it to its accepted value g = 9.81 m/s2.

 

Answer

 

So g = m/tanq and Dg/g =  (Dm/m + Dq/cos2q). Note Dq must be in radians!