Sheet 3 Potential Energy, Kinetic Energy and Mechanical Energy.
The potential energy of a certain spring when stretched through \(10cm\) is \(5J\). Calculate: the amount of work done so as to stretch the spring through an additional \(5cm\).
Ans: \(6.25J\).
A stone of mass \(0.4kg\) is thrown vertically upwards with a speed of \(9.8\frac{m}{s}\). What will be the kinetic energy and P.E after half second?
Ans: \(K.E=4.802J\) and \(P.E=14.386J\).
An object is attached to a vertical spring and slowly lowered to its equilibrium position. This stretches the spring by a distance \(l\). If, the same object is attached to the same vertical spring but permitted to fall instead, through what distance does it stretches the spring?
Ans: \(x=2l\).
The bob of a pendulum is released from a horizontal position A as shown in below figure.| If the length of the pendulum is \(1.5m\), what is the speed with which the bob arrives at the lower most position B ? Given: that it dissipates \(5\%\) of its initial energy against air resistance.
Ans: \(v=5.3\frac{m}{s}\).
A bullet of mass \(10g\) travels horizontally with speed of \(100\frac{m}{s}\) and is absorbed by a wooden block of mass \(900g\) suspended by a string. Calculate: the vertical height through which the block rises.(Given: \(g=10\frac{m}{s^{2}}\)).
Ans: \(h=5cm\).
In a ballistic demonstration, a police officer fires a bullet of mass \(50g\) with speed of \(200\frac{m}{s}\) on soft plywood of thickness \(2.00cm\). The bullet emerges only with \(10\%\) of its initial kinetic energy. What will be the emergent speed of the bullet?
Ans: \(v_{final}=63.2\frac{m}{s}\).
Show that the work done in raising a body through a vertical distance \(h\) up along a inclined plane is equal to \(mgh\).
The P.E between two atoms in a molecules is given by \(U(x)=\frac{a}{x^{12}}-\frac{b}{x^{6}}\), where a and b are positive constants and x is the distance between atoms. What is the separation between atoms at equilibrium?
A \(2000kg\) car starts to coast up the hill as shown in below figure. Its speed is \(20\frac{m}{s}\) at A. Its speed at point B is \(5\frac{m}{s}\). How large an average frictional force retarded its motion? Given: the distance along the road is \(40m\)
Ans: \(F_{friction}=4475N\).
Two masses as shown in below figure are released from their positions. Calculate: the velocity with which the mass \(5kg\) touches the ground if its initial height is from the surface of the earth is \(4m\). Also; show that the gain in kinetic energy of the system is equal to the loss in potential energy of the system. Take: \(g=10\frac{m}{s^{2}}\). Ans:
A \(4.0kg\) block has a speed of \(2\frac{m}{s}\) at point A and \(6\frac{m}{s}\) at point B. If the distance between point A and B along the curve is \(12m\) , how large a frictional force acts on it? Assuming the same friction how far from point B will it stop?
Ans: \(S=24.88m\).
The P.E of a \(1kg\) particle free to move along x-axis is given by \(U(x)=\left[\frac{x^{4}}{4}-\frac{x^{2}}{2}\right]J\). The total mechanical energy of the particle is \(2J\). Find: the maximum speed of particle.
Ans: \(v_{max}=\frac{3}{\sqrt{2}}\frac{m}{s}\).
Two spring has the spring constant \(K_{1}\) and \(K_{2}\), where \(K_{1}>K_{2}\). On which spring , more work is done if (i) They are stretched by the same force? (ii) They are stretched by the same amount?
Ans: (i) \(W_{2}>W_{1}\). and (ii) \(W_{1}>W_{2}\).
The horizontally placed spring has a force constant of \(48\frac{N}{m}\). The mass of the block attached to spring is \(8kg\) and the other end of the spring is fixed to a rigid vertical support. Initially the block is at rest and spring is un-stretched. The horizontal force of \(20N\) is applied on block then what is the speed of the block when it has been moved through a distance of \(0.5m\)?
Ans: \(v=1\frac{m}{s}\).
A spring gun has a spring constant of \(80\frac{N}{cm}\). The spring is compressed \(12cm\) by a ball of mass \(15g\). How much is the P.E of spring? If the trigger is pulled what will be the velocity of the ball?
Ans: \(U=57.6J\) and \(v=87.63\frac{m}{s}\).
To simulate the car accident, auto manufacturers study the collision of moving cars with horizontally mounted spring of spring constant \(\left(6.25\times10^{2}\right)\frac{N}{m}\). Consider a typical simulation with a car of mass \(1000kg\) moving with a speed of \(18.0\frac{km}{h}\) on a smooth horizontal road and colliding with the mentioned horizontal spring. What is the maximum compression of the spring ?
Ans: \(x_{max}=2m\).
A ball of mass \(m\) is dropped from a height \(h\) on a platform fixed at the top of a vertical spring as shown in the figure. The platform is depressed by a distance \(x\). What is the spring constant \(K\)?
Ans: \(K=\frac{2mg(h+x)}{x^{2}}\).
A particle moves in a straight line with its retardation proportional to its displacement. What will be the loss of kinetic energy for any displacement \(x\).
Ans: \(\Delta(K.E)=-\frac{mKx^{2}}{2}\).
A bolt of mass \(0.3kg\) falls from the celling of an elevator moving down with a uniform speed of \(7\frac{m}{s}\). It hits the floor of the elevator having length \(3m\) and does not rebounds. What is the amount of heat produced by the impact? Would your answer be different if the elevator is stationary?
Ans: \(E_{heat}=8.86J\) and Answer does not change .
A ball falls under gravity from a height \(h=10m\) with an initial speed of \(u\). If collides with the ground, looses half of its total energy and then rises back to the same height. Find: initial velocity of the ball.
Ans: \(u=14\frac{m}{s}\).
A ball at rest is dropped from a height of \(15m\). It looses \(30\%\) of its K.E, in striking the ground, find the height to which it bounces back?
Ans: \(h=10.5m\).
Consider a frictionless hemispherical bowl of radius \(R\). A ball of mass \(m\) is pushed down the wall from the point P. It just rises up the edge of the bowl. Calculate: the speed with which the ball is pushed down along wall.
Ans: \(v=\sqrt{2g(R-h)}\).
A mass less platform is placed on a light elastic spring. When a sand particle of mass \(0.1kg\) is dropped on the pan from a height of \(0.24m\), the particle strikes the pan and the spring compresses by \(0.01m\). From what height should the particle be dropped to cause a compression of \(0.04m\).
