Experiment: Circular Motion and Suspended Mass
Equipment: Stopwatch, string, washers, rulers, hollow plastic biro or straw
Method:Thread a string of about 60cm length through the straw. Tie a light washer of mass m onto one end of the string and tie a heavier washer of mass M onto the other end. Hold the set up by the straw, and you will notice that the heavier mass M pulls the light mass upwards (obviously). However, if you swing the light mass m in a circle of radius r and period T, then you can balance the heavy mass M and the light mass m. It takes practice, but give it a go.
When the two masses are balanced, what do you notice about the angle the string makes with the horizontal. Does it change as you change the length of the string L, the speed v, and the period T.
What is the relationship between the length L of the string in the spinning segment and the speed of rotation of the small mass m. If L is very small, you will need to spin the small mass m very fast to counteract the weight of the large mass M. If the length L is large, you don’t need to spin the small mass m as fast to balance the large mass M. Why?
What is the force balancing the mass M. As m whirls around, it experiences a centripetal force coming from the tension F in the string. As you whirl it faster, the tension F increases. And it is the tension in the string that is balancing the large mass M against its weight.
Introduction:Let’s analyse the Force balance equations. (Answers are given at the end of this document.)For the large mass M, write the vertical balance equations:
For the small mass m, write the vertical and horizontal force equations:
Use your relations to derive a formula for the angle made by the string when the small mass is rotating at linear speed v, with radius r and period T.
sinθ =
What does this imply for your setup with constant values of the small mass m and the large mass M. Test your derived relation for your equipment by measuring the angle θ, as well as the masses m and M. Work out the Percentage Error by comparing your measured sin θ with the ratio of the masses.
m = , M = , θ =, sin θ = m/M =
Percentage Error =
Using v = 2πr/T and both h=Lsinθ andr=Lcosθ, obtain a single formula relating T2 and L. (You will also need sinθ=m/M.)
Check the units on the left hand side of your equation are the same as the units on the right hand side.
Your relation means we can calculate the acceleration due to gravity. To average over the errors we will plot T2versusL to get a straight line and use the gradient to calculate g. Alternatively, we can use a quadratic line of best fit to calculate g.
Plotting T2 vs L gives a relation like y=mx+c. Write a formula for your expected Y axis intercept and the gradient of the line.
Intercept =
Gradient =
Plotting L vs T gives a relation like y=ax2+bx+c. Write a formula for your expected value of “a” and for the intercept with the Y axis.
Intercept =
Gradient =
Aim: To investigate the relationship between L and T and use this to measure the acceleration due to gravity.
Hypothesis: That T2 varies linearly with L, so a determination of the gradient allows a measurement of the acceleration due to gravity. And that L varies quadratically with T so quadratic regression gives a measurement of the acceleration due to gravity.
Data and Analysis: We will use a stop watch to measure the period of motion T (time at least 10 revolutions and divide the total time by the number of revolutions), as well as measuring the length of the string L as the washer swings in a circle (put your finger on the string and bring it to rest so you can measure the length L).
In this experiment, we will be using rulers and eye based measurements. If we wanted more accuracy, we would video the experiment and use a data analysis program like Tracker.
Complete the following data table using at least 10 readings. You first need m and M.
m = ; M = ;
L (m) / T (10 rev, s) / T (s) / T^2 (s^2)Use your data to make a plot of T2 vs h. Include a Linear Regression Trend Line with the equation displayed on the graph. From this equation, calculate a measured value of g.
Gradient =
Acceleration due to gravity g =
Now re-arrange your relation to make L the subject. Write a relation between L and T in the form y=ax2+bx+c. Use your data to make a plot of L vs T. Include a quadratic Regression Trend Line with the equation displayed on the graph. From this equation, calculate a measured value of g. The quadratic regression gives
a =
From this, calculate your value of the acceleration due to gravity g =
Percentage Error =
Conclusion: What do you conclude about circular motion and the acceleration due to gravity.
Answers:
F = Mg, Y: F sinθ=mg, X: F cosθ = m acent = m v2/r, sinθ =m/M=constant, T2=(4mπ2/Mg)L,
Intercept=0, Gradient=(4mπ2/Mg). I use M=2m so sinθ=1/2 and θ=30o.
Theoretical Data
T (s) / L (m) / T^2 (s^2)0.45345 / 0.1 / 0.205617
0.55536 / 0.15 / 0.308425
0.641275 / 0.2 / 0.411234
0.716967 / 0.25 / 0.514042
0.785398 / 0.3 / 0.61685
0.848327 / 0.35 / 0.719659
0.9069 / 0.4 / 0.822467
0.961912 / 0.45 / 0.925275
1.013945 / 0.5 / 1.028084
1.063434 / 0.55 / 1.130892
1.110721 / 0.6 / 1.233701
Linear Gradient =(4mπ2/Mg) = 2.0562 s2/m.
This rearranges to give acceleration
g=4mπ2/(M×4.0284)=9.6m/s2.
Quadratic equation is
L = (gM/4mπ2) T2.
Quadratic regression gives
0.4863=Mg/4mπ2 m/s/s.
This rearranges to give
g = 9.6m/s/s.