UIUC Physics 211 James Scholars Problem Set 2
In this assignment we will do some problems in 2-d motion and many-body systems in a bit more depth.
Consider a bead of mass m that is free to move around a horizontal, circular ring of wire (the wire passes through a hole in the bead). You may neglect gravity in this problem (assume the experiment is being done in space, far away from anything else). The radius of the ring of wire is r. The bead is given an initial speed v_0 and it slides with a coefficient of friction mu_k. In the subsequent steps we will investigate the motion at later times. You should begin by drawing a free-body diagram at some instant of time. Note that there will be a radial acceleration, a_R, and a tangential acceleration, a_T, in this problem.
1.1 Write Newton's 2nd law for the radial and tangential directions.
1.2 Combine the above equations to write a differential equation for dv/dt, where v is the speed at time t.
1.3 Solve the above differential equation to determine v(t). The solution has the form v = c1/(1+c2*t) - find c1 and c2. Hint: if v_0 = 3 m/s, mu_k = 0.1, r = 10 cm, and t = 3 s, v(3) = 0.3 m/s.
We will continue our analysis of Problem 1 with the bead.
2.1 Given your solution for v(t), calculate the radial and tangential components of the acceleration, a_R(t) and a_T(t), respectively. From these calculate the total acceleration a_tot(t).
2.2 Given your solution for v(t), write the differential equation involving ds/dt, where s is the position of the bead around the circumference of the ring. Solve this equation for s(t).
2.3 Graph s(t), v(t), a_R(t), and a_T(t) for v_0 = 3 m/s, mu_k = 0.1, r = 10 cm and let t vary from 0 to 10 s.
2.4 What is the speed, v_1, of the bead after 1 revolution of the ring (using the parameter values given in part 2.3)? What is the speed v_2 after 2 revolutions of the ring? (On your own you may wish to calculate these two speeds for the same bead with the same parameters, except moving along a straight wire.)
Consider the Atwood's machine of Lecture 8. We wish to use this machine to measure our local acceleration of gravity with an accuracy of 5% [i.e. (Delta g)/g = 0.05]. To begin, suppose we let the mass m_1 fall through a distance L.
3.1 Find an expression for the acceleration of gravity, g, in terms of m_1, m_2, L and t.
3.2 Now suppose we are able to measure time with an accuracy of (Delta t) = 0.1 s. Assuming that, for example, (Delta t)/t can be approximated by the differential dt/t, write the relationship between (Delta g)/g and (Delta t)/t. You can do this by starting with the derivative dg/dt determined from the equation in the previous part.
3.3 If L = 3 m and m_1 = 1 kg, determine the value of m_2 required to determine g to 5%. If we want to measure g to 1% would the mass m_2 increase or decrease - why? (On your own, you might want to consider the effect of the uncertainty in the masses of m_1 and m_2 on the determination of g.)