Two problems I am stuck on:
1. You have a pendulum made of a long thin brass rod with a mass on the end has an oscillation period of t0 at some temperature.
Show that the change in the period (delta t) of that pendulum is approximately delta t= (1/2)*alpha*t0*(delta T) when the temperature changes by an amount delta T (where alpha is the thermal expansion coefficient of the brass).
I know the period of a pendulum is 2*pi*sqrt(L/g) and I know when the temperature increases the formula for length increase is L=L(orig)*alpha*(delta T) but I cant quite seem to figure out how to put it all together.
2. Consider a flat plate of solid gold which has a circular hole in it. Suppose that a silver sphere slightly larger in diameter than the hole settles into the hole. The thermal expansion coefficients are alpha(gold)=13e-6 and alpha(silver)=17e-6.
If the ball and plate are always kept the same temperature, would that temperature have to increase or decrease fo rthe ball to fall through the hole? If the diameter of the ball were .1% larger than the diameter of the hole at room temperature, by how much would the temperature have to change for the ball to fall through? At what temperature would it fall through if you started at "room temperature"? Explain.
I know that it would have to get colder for it to fall through since the silver ball has a higher thermal expansion coefficient, it changes more in size with temperate than the gold. So if it got colder, the ball would shrink faster than the hole shrinks. But I have no idea how to figure out the last two questions.
1. You have a pendulum made of a long thin brass rod with a mass on the end has an oscillation period of t0 at some temperature.
Show that the change in the period (delta t) of that pendulum is approximately delta t= (1/2)*alpha*t0*(delta T) when the temperature changes by an amount delta T (where alpha is the thermal expansion coefficient of the brass).
I know the period of a pendulum is 2*pi*sqrt(L/g) and I know when the temperature increases the formula for length increase is L=L(orig)*alpha*(delta T) but I cant quite seem to figure out how to put it all together.
2. Consider a flat plate of solid gold which has a circular hole in it. Suppose that a silver sphere slightly larger in diameter than the hole settles into the hole. The thermal expansion coefficients are alpha(gold)=13e-6 and alpha(silver)=17e-6.
If the ball and plate are always kept the same temperature, would that temperature have to increase or decrease fo rthe ball to fall through the hole? If the diameter of the ball were .1% larger than the diameter of the hole at room temperature, by how much would the temperature have to change for the ball to fall through? At what temperature would it fall through if you started at "room temperature"? Explain.
I know that it would have to get colder for it to fall through since the silver ball has a higher thermal expansion coefficient, it changes more in size with temperate than the gold. So if it got colder, the ball would shrink faster than the hole shrinks. But I have no idea how to figure out the last two questions.
