Work, Energy and Power
- The kinetic energies of two similar cars A and B are 100 J and 225 J respectively. On applying breaks, car A stops after 1000 m and car B stops after 1500 m. If F_A and F_B are the forces applied by the breaks on cars A and B, respectively, then the… Answer: 2. 2/3 Key idea: by the work-energy theorem, the brake force does work equal to the kinetic energy lost: F d = K . F_A=100/1000=0.1\... 1
- A bob of heavy mass m is suspended by a light string of length l. The bob is given a horizontal velocity v_0 as shown in figure. If the string gets slack at some point P making an angle θ from the horizontal, the ratio of the speed v of the bob at po… Answer: 4. (sinθ/2+3sinθ)^(1/2) Key idea: the string goes slack when the tension becomes zero, and energy is conserved along the way. At P (... 1
- Two bodies A and B of same mass undergo completely inelastic one dimensional collision. The body A moves with velocity v_1 while body B is at rest before collision. The velocity of the system after collision is v_2. The ratio v_1 : v_2 is : Answer: 3. 2:1 Key idea: In a completely inelastic collision the bodies stick together and momentum is conserved. mv₁+0=(m+m)v₂ , so v₂=v₁/2... 1
- At any instant of time t, the displacement of any particle is given by 2t-1 (SI unit) under the influence of force of 5N. The value of instantaneous power is (in SI unit): Answer: 2. 10 Key idea: Instantaneous power P=vec F·vec v . Velocity v=d/dt(2t-1)=2 m/s (constant). Then P=Fv=5×2=10 W. Option 2. 1
- The potential energy of a long spring when stretched by 2 cm is U. If the spring is stretched by 8 cm, potential energy stored in it will be : Answer: 1. 16U Spring energy U=1/2kx²∝ x² . Extension goes from 2 cm to 8 cm, a factor of 4, so energy goes up by 4²=16 . The new energy is... 1
- An electric lift with a maximum load of 2000 kg (lift + passengers) is moving up with a constant speed of 1.5 ms^-1. The frictional force opposing the motion is 3000 N. The minimum power delivered by the motor to the lift in watts is : (g = 10 ms^-2) Answer: 4. 34500 At constant speed the motor force balances weight plus friction. F=mg+f=2000×10+3000=23000\ N P=Fv=23000×1.5=34500\ W Optio... 1
- Water falls from a height of 60 m at the rate of 15 kg/s to operate a turbine. The losses due to frictional force are 10% of the input energy. How much power is generated by the turbine ? (g = 10 m/s^2) Answer: 2. 8.1 kW Key idea: power input = (mass per second) × g × h; the turbine gets 90% of it. P_in=15×10×60=9000\ mathrm W P_out=0.9×9000... 1
- A particle is released from height S from the surface of the Earth. At a certain height its kinetic energy is three times its potential energy. The height from the surface of earth and the speed of the particle at that instant are respectively : Answer: 4. S/4, √(3gS/2) Key idea: total mechanical energy mgS is conserved (taking the surface as zero of potential energy). At height h: U... 1
- A force F = 20 + 10y acts on a particle in y-direction where F is in newton and y in meter. Work done by this force to move the particle from y = 0 to y = 1 m is : Answer: 3. 25 J Work done by a variable force is W=∫ F dy . W=∫₀¹(20+10y) dy=[20y+5y²]₀¹=20+5=25 J. 1
- Body A of mass 4m moving with speed u collides with another body B of mass 2m, at rest. The collision is head on and elastic in nature. After the collision the fraction of energy lost by the colliding body A is : Answer: 2. 8/9 Use the formula for the final speed of a body in a head-on elastic collision with a body at rest. v_A=m₁-m₂/m₁+m₂u=4m-2m/4m+2... 1
- A mass m is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break when : Answer: 3. the mass is at the lowest point The wire breaks where its tension is largest. At the lowest point: T-mg=mv²/r , so T=mg+mv²/r . H... 1
- A moving block having mass m, collides with another stationary block having mass 4m. The lighter block comes to rest after collision. When the initial velocity of the lighter block is v, then the value of coefficient of restitution (e) will be Answer: 3. 0.25 Momentum conservation: mv=0+4mv₂⇒ v₂=v/4 (lighter block at rest, v₁=0 ). Coefficient of restitution = relative speed of sepa... 1
- A body initially at rest and sliding along a frictionless track from a height h (as shown in the figure) just completes a vertical circle of diameter AB = D. The height h is equal to Answer: 4. 5/4 D To just complete the vertical circle of radius R=D/2 , the speed at the top B must satisfy v_B²=gR . Energy conservation fr... 1
- A toy car with charge q moves on a frictionless horizontal plane surface under the influence of a uniform electric field E. Due to the force q E, its velocity increases from 0 to 6 m/s in one second duration. At that instant the direction of the fiel… Answer: 3. 1 m/s, 3 m/s Phase 1 (0 to 1 s): a=6\ m/s² , displacement =1/2·6·1=3 m, ending at 6 m/s. Phase 2 (1 to 3 s): the acceleration rev... 1
- Consider a drop of rain water having mass 1 g falling from a height of 1 km. It hits the ground with a speed of 50 m/s. Take 'g' constant with a value 10 m/s^2. The work done by the (i) gravitational force and the (ii) resistive force of air is : Answer: 3. (i) 10 J (ii) - 8.75 J Key idea: use work-energy theorem: W_g+W_air=Δ KE . (i) W_g=mgh=0.001×10×1000=10 J. Δ KE=1/2(0.001)(50)²=1... 1
- A bullet of mass 10 g moving horizontally with a velocity of 400 m s^-1 strikes a wooden block of mass 2 kg which is suspended by a light inextensible string of length 5 m. As a result, the centre of gravity of the block is found to rise a vertical d… Answer: 2. 120 m s ⁻¹ Use energy for the swing of the block and momentum conservation for the collision. Block after impact: v=√(2gh)=√(2×10... 1
- Two identical balls A and B having velocities of 0.5 m/s and -0.3 m/s respectively collide elastically in one dimension. The velocities of B and A after the collision respectively will be Answer: 1. 0.5 m/s and -0.3 m/s For an elastic head-on collision between two identical masses, the bodies simply exchange velocities. (Momen... 1
- A particle moves from a point (-2 i+5 j) to (4 j+3 k) when a force of (4 i+3 j) N is applied. How much work has been done by the force? Answer: 2. 5 J Work =vec F·vec d , where vec d is the displacement. vec d=(4hat j+3hat k)-(-2hat i+5hat j)=2hat i-hat j+3hat k . W=(4hat i+3... 1
- A body of mass 1 kg begins to move under the action of a time dependent force F = (2t i + 3t^2 j) N, where i and j are unit vectors along x and y axis. What power will be developed by the force at the time t ? Answer: 4. (2t³ + 3t⁵) W Key idea: power P=vec F·vec v ; find vec v by integrating vec a=vec F/m . vec a=2that i+3t²hat j (m = 1 kg), so vec... 1
- What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop ? Answer: 4. √(5gR) Key idea: at the top the minimum speed satisfies mg=mv_t²/R , so v_t²=gR . Energy conservation from bottom to top (height... 1