Question

A bead is released from rest at point P and slides along a wire, as shown. The track loop around and forms a vertical circle is formed of diameter 40cm. At point Q the bead has a speed $1.4m{{s}^{-1}}$. The air resistance and friction on the wire are negligible. What is the height h from which the bead is at the initial point?

A. 0.30m
B. 0.40m
C. 0.50m
D. 0.60m

Answer 1

Whether an object is at rest or in motion the total energy of the object will always remain the same. The Law of Conservation of Energy states that energy cannot be destroyed or created and it can only be transferred from one form to another. Every object in the universe obeys the conservation Law of energy.

Formula used:

& E={{K}_{E}}+{{P}_{E}} \\\ & {{K}_{E}}=\dfrac{1}{2}m{{v}^{2}} \\\ & {{P}_{E}}=mgh \\\ \end{aligned}$$ **Complete step by step answer:** Using the concept of law of conservation of energy which states that, an energy cannot be created nor destroyed it can only be transferred from one form to another form of energy. The energy at point P will be the same as total energy at point Q which is 40cm above the reference level. Mathematically, $\begin{aligned} & {{E}_{p}}={{E}_{Q}} \\\ & {{K}_{Ep}}+{{P}_{EP}}={{K}_{EQ}}+{{P}_{EQ}} \\\ \end{aligned}$ Where, K and P stands for kinetic and potential energies respectively. By putting the value in the formula as per the conditions provided $0+mgh=\dfrac{1}{2}m{{(1.4)}^{2}}+mg(0.4)$ For easy calculation, let us consider the value of acceleration due to gravity (g) be equal to 10 $\begin{aligned} & 10mh=\dfrac{1.96}{2}m+4m \\\ & \Rightarrow 10h=\dfrac{1.96}{2}+4 \\\ & \Rightarrow h=\dfrac{4.98}{10} \\\ & \therefore h=0.498m \\\ \end{aligned}$ The calculated value is approximately equivalent to $0.5m$ Thus we can conclude that the correct option which shows the right value of h and satisfies all the conditions mentioned in the question is option C. **Note:** The energy comprises two components. One is kinetic energy and the second is potential energy. Potential energy is also known as stored energy. The potential energy of an object is very dependent on the position of the object from the reference point. The kinetic energy is the energy used by an object for motion or displacement. When we assign values in any equation they must be in the same unit standards.