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Question: If $\operatorname{Cos} X=\frac{-1}{2}$, find the value of $\cos 3 x$ Solution: Given: $\operatorname{Cos} X=\frac{-1}{2}$ To find: $\cos 3 x$ We know that, $\cos 3 x=4 \cos ^{3} x-3 \cos x$ Putting the values, we get $\cos 3 x=4 \times\left(-\frac{1}{2}\right)^{3}-3 \times\left(-\frac{1}{2}\right)$ $\cos 3 x=4 \times\left(-\frac{1}{8}\right)+\frac{3}{2}$ $\cos 3 x=\left[-\frac{1}{2}+\frac{3}{2}\right]$ $\cos 3 x=\left[\frac{-1+3}{2}\right]$ $\cos 3 x=\frac{2}{2}$ $\cos 3 x=1$...
Read More →An engine is attached to a wagon through
Question: 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. Solution: KE = 1/2 mv2 m = 50000 kg v = 10 m/s KE = 2500000J KE of spring = 10% of the KE...
Read More →Suppose the average mass of raindrops
Question: Suppose the average mass of raindrops is 3.0 10-5kg 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. Solution: 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= 4.05104J...
Read More →A raindrop of mass 1.00 g falling from
Question: 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? Solution: (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...
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Question: If $\operatorname{Sin} X=\frac{1}{6}$, find the value of $\sin 3 x$ Solution: $\operatorname{Sin} X=\frac{1}{6}$ Given: $\operatorname{Sin} X=\frac{1}{6}$ To find: $\sin 3 x$ We know that, $\sin 3 x=3 \sin x-\sin ^{3} x$ Putting the values, we get $\sin 3 x=3 \times\left(\frac{1}{6}\right)-\left(\frac{1}{6}\right)^{3}$ $\sin 3 x=\frac{1}{6}\left[3-\left(\frac{1}{6}\right)^{2}\right]$ $\sin 3 x=\frac{1}{6}\left[3-\frac{1}{36}\right]$ $\sin 3 x=\frac{1}{6}\left[\frac{108-1}{36}\right]$ $...
Read More →Consider a one-dimensional motion
Question: 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. Solution: For region A, E = V + K and V E which means that the KE is negative and this is not possible. For region B, K = E V and V...
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Question: If $\tan \mathrm{x}=\frac{-5}{12}$ and $\frac{\pi}{2}\mathrm{x}\pi$ find the values of tan 2x Solution: Given: $\tan \mathrm{x}=-\frac{5}{12}$ To find: $\tan 2 x$ We know that, $\tan 2 x=\frac{2 \tan x}{1-\tan ^{2} x}$ Putting the values, we get $\tan 2 x=\frac{2 \times\left(-\frac{5}{12}\right)}{1-\left(-\frac{5}{12}\right)^{2}}$ $\tan 2 x=\frac{-\frac{5}{6}}{1-\frac{25}{144}}$ $\tan 2 x=\frac{-5}{6\left(\frac{144-25}{144}\right)}$ $\tan 2 x=\frac{-5 \times 144}{6 \times 119}$ $\tan 2...
Read More →Two bodies of unequal mass are moving
Question: 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? Solution: KE1 = KE2 WD1 = WD2 F1s1 = F2s2 F1 = F2 s1 = s2...
Read More →Give an example of a situation in
Question: Give an example of a situation in which an applied force does not result in a change in kinetic energy. Solution: Work done during the circular motion there is no change in the kinetic energy....
Read More →The average work done by a human heart
Question: 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. Solution: P = WD/time WD is one beat of heart = 0.5 J WD in 72 beats = 36 J P = WD/t = 0.6 W...
Read More →Calculate the power of a crane in watts,
Question: Calculate the power of a crane in watts, which lifts a mass of 100 kg to a height of 10 min 20 sec. Solution: P = WD/time = Fs cos /t = mgh cos /t h = 10 m t = 20 sec F = mg = 1000 Therefore, P = 500 Watts...
Read More →Solve this
Question: If $\tan \mathrm{x}=\frac{-5}{12}$ and $\frac{\pi}{2}\mathrm{x}\pi$ find the values of cos 2x Solution: Given: $\tan \mathrm{x}=-\frac{5}{12}$ To find: cos 2x We know that, $\cos 2 x=\frac{1-\tan ^{2} x}{1+\tan ^{2} x}$ Putting the values, we get $\cos 2 x=\frac{1-\left(-\frac{5}{12}\right)^{2}}{1+\left(-\frac{5}{12}\right)^{2}}$ $\cos 2 x=\frac{1-\frac{25}{144}}{1+\frac{25}{144}}$ $\cos 2 x=\frac{\frac{144-25}{144}}{\left(\frac{144+25}{144}\right)}$ $\cos 2 x=\frac{\frac{119}{144}}{\f...
Read More →In an elastic collision of two billiard balls,
Question: 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. Solution: The kinetic energy and the total linear momentum of the billiard balls are conserved as there is no non-conservative force....
Read More →A body is moved along a closed loop.
Question: 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. Solution: The work done by the moving body is zero when the conservative force is acting on the body during the motion. The work done by the moving body is non-zero when the non-conservative force is acting on the body....
Read More →A body falls towards earth in the air.
Question: A body falls towards earth in the air. Will its total mechanical energy be conserved during the fall? Justify. Solution: The total mechanical energy of the free-falling body is no conserved as this energy is used against the force of friction from the air molecules....
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Question: If $\tan x=\frac{-5}{12}$ and $\frac{\pi}{2}x\pi$find the values of sin 2x Solution: Given: $\tan x=-\frac{5}{12}$ To find: sin 2x We know that, $\sin 2 x=\frac{2 \tan x}{1+\tan ^{2} x}$ Putting the values, we get $\sin 2 x=\frac{2 \times\left(-\frac{5}{12}\right)}{1+\left(-\frac{5}{12}\right)^{2}}$ $\sin 2 x=\frac{-\frac{5}{6}}{1+\frac{25}{144}}$ $\sin 2 x=\frac{-5}{6\left(\frac{144+25}{144}\right)}$ $\sin 2 x=\frac{-5 \times 144}{6 \times 169}$ $\sin 2 x=\frac{-5 \times 24}{169}$ $\s...
Read More →Calculate the work done by a car against
Question: 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. Solution: WD = Fs cos WD = Fs cos 90o= 0 Therefore, the work done by the car against the gravity is zero....
Read More →A body is being raised to a height h from
Question: 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 Solution: (a) The work done by the applied force is positive (b) The work done by the gravitational force is negative...
Read More →Why is electrical power required at all
Question: 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? Solution: There is a limit on the number of passengers in case of an elevator descending as it is not a free fall and moves down with a uniform speed....
Read More →Verify Rolle's theorem for each of the following functions on the indicated intervals
Question: Verify Rolle's theorem for each of the following functions on the indicated intervals (i) $f(x)=\cos 2(x-\pi / 4)$ on $[0, \pi / 2]$ (ii) $f(x)=\sin 2 x$ on $[0, \pi / 2]$ (iii) $f(x)=\cos 2 x$ on $[-\pi / 4, \pi / 4]$ (iv) $f(x)=e^{x} \sin x$ on $[0, \pi]$ (v) $f(x)=e^{x} \cos x$ on $[-\pi / 2, \pi / 2]$ (vi) $f(x)=\cos 2 x$ on $[0, \pi]$ (vii) $f(x)=\frac{\sin x}{e^{x}}$ on $0 \leq x \leq \pi$ (viii) $f(x)=\sin 3 x$ on $[0, \pi]$ (ix) $f(x)=e^{1-x^{2}}$ on $[-1,1]$ (x) $f(x)=\log \le...
Read More →Verify Rolle's theorem for each of the following functions on the indicated intervals
Question: Verify Rolle's theorem for each of the following functions on the indicated intervals (i) $f(x)=\cos 2(x-\pi / 4)$ on $[0, \pi / 2]$ (ii) $f(x)=\sin 2 x$ on $[0, \pi / 2]$ (iii) $f(x)=\cos 2 x$ on $[-\pi / 4, \pi / 4]$ (iv) $f(x)=e^{x} \sin x$ on $[0, \pi]$ (v) $f(x)=e^{x} \cos x$ on $[-\pi / 2, \pi / 2]$ (vi) $f(x)=\cos 2 x$ on $[0, \pi]$ (vii) $f(x)=\frac{\sin x}{e^{x}}$ on $0 \leq x \leq \pi$ (viii) $f(x)=\sin 3 x$ on $[0, \pi]$ (ix) $f(x)=e^{1-x^{2}}$ on $[-1,1]$ (x) $f(x)=\log \le...
Read More →A bullet of mass m fired at 30o to the horizontal
Question: 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...
Read More →A man, of mass m, standing at the bottom of the staircase,
Question: 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 Solution: (b) work done by all the force on man is zero (d) the reaction force from a step does not do wor...
Read More →A cricket ball of mass 150 g moving with a speed of 126 km/h
Question: 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 104N (d) 2.1 104N Solution: (c) 1.05 104N...
Read More →Verify Rolle's theorem for each of the following functions on the indicated intervals
Question: Verify Rolle's theorem for each of the following functions on the indicated intervals (i) $f(x)=\cos 2(x-\pi / 4)$ on $[0, \pi / 2]$ (ii) $f(x)=\sin 2 x$ on $[0, \pi / 2]$ (iii) $f(x)=\cos 2 x$ on $[-\pi / 4, \pi / 4]$ (iv) $f(x)=e^{x} \sin x$ on $[0, \pi]$ (v) $f(x)=e^{x} \cos x$ on $[-\pi / 2, \pi / 2]$ (vi) $f(x)=\cos 2 x$ on $[0, \pi]$ (vii) $f(x)=\frac{\sin x}{e^{x}}$ on $0 \leq x \leq \pi$ (viii) $f(x)=\sin 3 x$ on $[0, \pi]$ (ix) $f(x)=e^{1-x^{2}}$ on $[-1,1]$ (x) $f(x)=\log \le...
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