The net buoyant force Equals vpg when the dragging viscous force of the air on the balloon is ignored. Where v is the volume of air that has been displaced. The upward net density is denoted by p....
Two identical steel cubes collide head-on face to face with a speed of 10 cm/s each. Find the maximum compression of each. Young’s modulus for steel = Y = 2 × 1011 N/m2.
Y = stress/strain Y = FL/A∆L WD = F∆L KE = 5 × 10-4 J WD = KE ∆L = 5 × 10-7 m
A rocket accelerates straight up by ejecting gas downwards. In a small time interval ∆t, it ejects a gas of mass ∆m at a relative speed u. Calculate KE of the entire system at t + ∆t and t and show that the device that ejects gas does work = (1/2) ∆m u2 in this time interval.
M is the rocket's mass at any given moment t. The rocket's velocity is v. The mass of the gas expelled during the time interval t is m. As a result, K = 1/2 u2∆m
A curved surface as shown in the figure. The portion BCD is free of friction. There are three spherical balls of identical radii and masses. Balls are released from one by one from A which is at a slightly greater height than C. with the surface AB, ball 1 has large enough friction to cause rolling down without slipping; ball 2 has a small friction and ball 3 has a negligible friction. a) for which balls is total mechanical energy conserved? b) which ball can reach D? c) for balls which do not reach D, which of the balls can reach back A?
a) For ball 1 the total mechanical energy is conserved b) Ball 1 reaches D c) Ball 3 reaches back A
A block of mass 1 kg is pushed up a surface inclined to horizontal at an angle of 30o by a force of 10 N parallel to the inclined surface. The coefficient of friction between the block and the incline is 0.1. If the block is pushed up by 10 m along the incline, calculate a) work done against gravity b) work done against the force of friction c) increase in potential energy d) increase in kinetic energy e) work done by an applied force
a) Work against gravity equals mgh 5 m= h 50 J WD against gravity b) The work done against the friction force is fs = 53 J. d) WD against gravity = 50 J increase in PE d) The system's increase in KE...
On complete combustion, a litre of petrol gives off heat equivalent to 3 × 107 J. In a test drive a car weighing 1200 kg, including the mass of driver, runs 15 km per litre while moving with a uniform speed on a surface and air to be uniform, calculate the force of friction acting on the car during the test drive, if the efficiency of the car engine were 0.5.
car engine Efficiency = 0.5 Energy given by the car with 1 litre of petrol = 1.5 × 107 WD = 1.5 × 107 f = 103 N
An adult weighing 600 N raises the centre of gravity of his body by 0.25 m while taking each step of 1 m length in jogging. If he jogs for 6 km, calculate the energy utilized by him in jogging assuming that there is no energy loss due to friction of ground and air. Assuming that the body of the adult is capable of converting 10% of energy intake in the form of food, calculate the energy equivalents of food that would be required to compensate energy utilized for jogging.
The energy used up is given as = mgh mg = 600 N h = 0.25m No.of steps in 6 km = 6000 steps Energy used in 6000 m = (6000)(600)(0.25)J Energy utilized in jogging = 9 × 104 J
An engine is attached to a wagon through a shock absorber of length 1.5 m. The system with a total mass of 50,000 kg is moving with a speed of 36 km/h when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by 1.0 m. If 90% of the energy of the wagon is lost due to friction, calculate the spring constant.
KE = 1/2 mv2 m = 50000 kg v = 10 m/s KE = 2500000J KE of spring = 10% of the KE wagon K = 5 × 105 N/m
Suppose the average mass of raindrops is 3.0 × 10^(-5) kg and their average terminal velocity 9 m/s. Calculate the energy transferred by rain to each square meter of the surface at a place which receives 100 cm of rain in a year.
Energy transferred by the rain to the surface of the earth = 1/2 mv2 The velocity of the rain = 9 m/s Mass = (volume)(density) = 1000 kg Energy transferred by 100 cm rainfall = 1/2 mv2 =...
Two pendulums with identical bobs and lengths are suspended from a common support such that in rest position the two bobs are in common. One of the bobs is released after being displaced by 10o so that it collides elastically head-on with the other bob. a) describe the motion of two bobs b) draw a graph showing variation in energy of either pendulum with time for 0 ≤ t ≤ 2T where T is the period of each pendulum
A raindrop of mass 1.00 g falling from a height of 1 km hits the ground with a speed of 50 m/s. Calculate a) the loss of PE of the drop b) the gain in KE of the drop c) is the gain in KE equal to loss of PE? If not why?
a) PE at the highest point = 10 J b) Gain in KE = 1/2 mv2 = 1.250 J c) Gain in KE is not equal to the PE
The bob A of a pendulum released from horizontal to the vertical hits another bob B of the same mass at rest on a table as shown in the figure. If the length of the pendulum is 1 m, calculate a) the height to which bob A will rise after collision b) the speed with which bob B starts moving. Neglect the size of the bobs and assume the collision to be elastic.
a) After the impact, bob A does not rise much because the PE of bob A is converted to KE and the momentum is transferred to bob B. (B) The speed of bob B is calculated as the sum of bob A's KE and...
Consider a one-dimensional motion of a particle with total energy E. There are four regions A, B, C, and D in which the relation between potential energy V, kinetic energy (K) and total energy is as given below: Region A: V > E Region B: V < E Region C: K > E Region D: V > K State with reason in each case whether a particle can be found in the given region or not.
E = V + K and V > E for area A, implying that the KE is negative and therefore this is not feasible. K = E – V and V E for area B, implying that both energies are larger than zero. V = E – K and...
A ball of mass m, moving with a speed 2v0 collides inelastically with an identical ball at rest. Show that a) for a head-on collision, both the balls move forward b) for a general collision, the angle between the two velocities of scattered balls is less than 90o.
a) Let v1 and v2 be the velocities of the two balls after the collision. According to the law of conservation of momentum, mv0 = mv1 + mv2 v2 = v1 + 2ev0 e < 1 b) Using the law of conservation of...
A graph of potential energy V(x) versus x is shown in the figure. A particle of energy E0 is executing motion in it. Draw graph of velocity and kinetic energy versus x for one complete cycle AFA.
From the given graph of KE versus x From the below graph of velocity versus x
A bob of mass m suspended by a light string of length L is whirled into a vertical circle as shown in the figure. What will be the trajectory of the particle if the string is cut at a) point B b) point C c) point X
a) When the string is severed at point B, the particle's tangential velocity will be vertically downward, and the bob will travel in the same direction. b) When the string is severed at point C, the...
Two bodies of unequal mass are moving in the same direction with equal kinetic energy. The two bodies are brought to rest by applying retarding force of the same magnitude. How would the distance moved by them before coming to rest compare?
KE1 = KE2 WD1 = WD2 F1s1 = F2s2 F1 = F2 s1 = s2
Give an example of a situation in which an applied force does not result in a change in kinetic energy.
The kinetic energy of work done in a circular motion remains unchanged.
The average work done by a human heart while it beats once is 0.5 J. Calculate the power used by heart if it beats 72 times in a minute.
P = WD/time WD is one beat of heart = 0.5 J WD in 72 beats = 36 J P = WD/t = 0.6 W
Calculate the power of a crane in watts, which lifts a mass of 100 kg to a height of 10 min 20 sec.
P = WD/time = Fs cos θ/t = mgh cos θ/t h = 10 m t = 20 sec F = mg = 1000 Therefore, P = 500 Watts
In an elastic collision of two billiard balls, which of the following quantities remain conserved during the short time of collision of the balls a) kinetic energy b) total linear momentum Give a reason for your answer in each case.
Because there is no non-conservative force, the kinetic energy and total linear momentum of the billiard balls are preserved.
A body is moved along a closed loop. Is the work done in moving the body necessarily zero? If not, state the condition under which work done over a closed path is always zero.
When the conservative force acts on the body during motion, the work done by the moving body is zero. When a non-conservative force acts on a moving body, the work done by the body is not zero.
A body falls towards earth in the air. Will its total mechanical energy be conserved during the fall? Justify.
The free-falling body's total mechanical energy is not preserved since it is utilised to overcome the frictional force of the air molecules.
Calculate the work done by a car against gravity in moving along a straight horizontal road. The mass of the car is 400 kg and the distance moved is 2m.
WD = Fs cos θ WD = Fs cos 90o = 0 Hence, the work done by the car against the gravity is zero.
A body is being raised to a height h from the surface of the earth. What is the sign of work done by a) applied force b) gravitational force
a) The applied force produces positive work. b) The gravitational pull produces negative work.
Why is electrical power required at all when the elevator is descending? Why should there be a limit on the number of passengers in this case?
In the event of an elevator falling, the number of passengers is limited because it is not a free fall and descends at a constant speed.
A rough inclined plane is placed on a cart moving with a constant velocity u on horizontal ground. A block of mass M rests on the incline. Is any work done by a force of friction between the block and incline? Is there then a dissipation of energy?
The block is clearly tilted on the plane in the illustration above. There is no work done since there is no displacement and no waste of energy.
Two blocks M1 and M2 having equal mass are free to move on a horizontal frictionless surface. M2 is attached to a massless spring as shown in the figure. Initially, M2 is at rest and M1 is moving toward M2 with speed v and collides head-on with M2.;a) while spring is fully compressed all the KE of M1 is stored as PE of spring b) while spring is fully compressed the system momentum is not conserved, though final momentum is equal to the initial momentum c) if spring is massless, the final state of the M1 is the state of rest d) if the surface on which blocks are moving has friction, then a collision cannot be elastic
c) if spring is massless, the final state of the M1 is a state of rest d) if the surface on which blocks are moving has friction, then a collision cannot be elastic
A bullet of mass m fired at 30o to the horizontal leaves the barrel of the gun with a velocity v. The bullet hits a soft target at a height h above the ground while it is moving downward and emerges out with half the kinetic energy it had before hitting the target. Which of the following statements are correct in respect of bullet after it emerges out of the target? a) the velocity of the bullet will be reduced to half its initial value b) the velocity of the bullet will be more than half of its earlier velocity c) the bullet will continue to move along the same parabolic path d) the bullet will move in a different parabolic path e) the bullet will fall vertically downward after hitting the target f) the internal energy of the particles of the target will increase
b) the velocity of the bullet will be more than half of its earlier velocity d) the bullet will move in a different parabolic path f) the internal energy of the particles of the target will...
A man, of mass m, standing at the bottom of the staircase, of height L, climbs it and stands at its top. a) work done by all forces on man is zero b) work done by all the force on man is zero c) work done by the gravitational force on man is mgL d) the reaction force from a step does not do work because the point of application of the force does not move while the force exists
b) work done by all the force on man is zero d) the reaction force from a step does not do work because the point of application of the force does not move while the force exists
A cricket ball of mass 150 g moving with a speed of 126 km/h hits at the middle of the bat, held firmly at its position by the batsman. The ball moves straight back to the bowler after hitting the bat. Assuming that collision between ball and bat is completely elastic and the two remain in contact for 0.001 sec, the force that the batsman had to apply to hold the bat firmly at its place would be a) 10.5 N b) 21 N c) 1.05 × ${10}^{4}$ N d) $2.1 \times {10}^{4}$N
c) 1.05 × 104 N
Which of the diagrams in figure correctly shows the change in kinetic energy of an iron sphere falling freely in a lake having sufficient depth to impart it a terminal velocity?
(b
In a shotput event an athlete throws the shotput of mass 10 kg with an initial speed of 1 m/s at 45o from height 1.5 m above ground. Assuming air resistance to be negligible and acceleration due to gravity to be 10 m/s2, the kinetic energy of the shotput when it just reaches the ground will be a) 2.5 J b) 5.0 J c) 52.5 J d) 155.0 J
d) 155.0 J
A mass of 5 kg is moving along a circular path of radius 1 m. If the mass moves with 300 revolutions per minute, its kinetic energy would be a) 250π2 b) 100 π2 c) 5 π2 d) 0
a) 250π2
Which of the diagrams shown in the figure represents variations of total mechanical energy of a pendulum oscillating in air as a function of time?
c)
Which of the diagrams shown in figure most closely shows the variation in kinetic energy of the earth as it moves once around the sun in its elliptical orbit?
d)
A body is moving unidirectionally under the influence of a source of constant power supplying energy. Which of the diagrams shown in the figure correctly shows the displacement-time curve for its motion?
b)
A body of mass 0.5 kg travels in a straight line with velocity v = ax3/2 where a = 5 m-1/2s-1. The work done by the net force during its displacement from x = 0 to x = 2 m is a) 1.5 J b) 50 J c) 10 J d) 100 J
b) 50 J
The potential energy function for a particle executing linear SHM is given by where k is the force constant of the oscillator. For k = 0.5 N/m, the graph of V(x) versus x is shown in the figure. A particle of total energy E turns back when it reaches x = ±xm, then which of the following is correct?a) V = O, K = E b) V = E, K = O c) V < E, K = O d) V = O, K < E
b) V = E, K = O
Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide down from rest, one on each track as shown in the figure. Which of the following statement is correct?a) both the stones reach the bottom at the same time but not with the same speed b) both stone reach the bottom with the same speed and stone I reaches the bottom earlier than stone II c) both the stones reach the bottom with the same speed and stone II reaches the bottom earlier than stone I d) both the stones reach the bottom at different times and with different speeds
c) both the stones reach the bottom with the same speed and stone II reaches the bottom earlier than stone I
During inelastic collision between two bodies, which of the following quantities always remain conserved? a) total kinetic energy b) total mechanical energy c) total linear momentum d) speed of each body
c) total linear momentum
A body is falling freely under the action of gravity alone in a vacuum. Which of the following quantities remain constant during the fall? a) kinetic energy b) potential energy c) total mechanical energy d) total linear momentum
c) total mechanical energy
A bicyclist comes to a skidding stop in 10 m. During this process, the force on the bicyclist due to the road is 200 N and is directly opposed to the motion. The work done by the cycle on the road is a) +2000 J b) -200 J c) zero d) -20,000 J
c) zero
A man squatting on the ground gets straight up and stand. The force of reaction of ground on the man during the process is a) constant and equal to mg in magnitude b) constant and greater than mg in magnitude c) variable but always greater than mg d) at first greater than mg, and later becomes equal to mg
d) at first greater than mg, and later becomes equal to mg
A proton is kept at rest. A positively charged particle is released from rest at a distance d in its field. Consider two experiments; one in which the charged particles is also a proton and in another, a positron. In the same time t, the work done on the two moving charged particles is a) the same as the same force law is involved in the two experiments b) less for the case of a positron, as the positron moves away more rapidly and the force on it weakens c) more for the case of a positron, as the positron moves away from a larger distance d) same as the work done by charged particle on the stationary proton
c) more for the case of a positron, as the positron moves away from a larger distance
An electron and a proton are moving under the influence of mutual forces. In calculating the change in the kinetic energy of the system during motion, one ignores the magnetic force of one on another. This is because, a) the two magnetic forces are equal and opposite, so they produce no net effect b) the magnetic forces do no work on each particle c) the magnetic forces do equal and opposite work on each particle d) the magnetic forces are necessarily negligible
b) the magnetic forces do no work on each particle