355847
A particle is moved from \((0,0)\) to \((2\;m,2\;m)\) with force \(\vec{F}=(x \hat{j}) N\) along path (i) \(y = x\) (ii) \(y=\sqrt{x}\). The work done about two paths is:
1 \(2\;J,2\;J\)
2 \(2J,\frac{8}{3}J\)
3 \(\frac{8}{3}J,\frac{8}{3}J\)
4 \(9\;J,8\;J\)
Explanation:
\(d W=\vec{F} \cdot d \vec{s}=(x \hat{j}) \cdot(d x \hat{i}+d y \hat{j})=x d y\) (i) when \(y=x\) \(W=\int d W=\int_{0}^{2} y d y=\left|\dfrac{y^{2}}{2}\right|_{0}^{2}=2 J\) (ii) When \(y=\sqrt{x}\) or \(y^{2}=x\) \(W=\int_{0}^{2} y^{2} d y=\left|\dfrac{y^{3}}{3}\right|_{0}^{2}=\dfrac{8}{3} J\)
PHXI06:WORK ENERGY AND POWER
355848
A chain of linear mass density 3 \(kg/m\) and length \(8 m\) is lying on the table with 4\(m\) of chain hanging from the edge. The work done in lifting the chain on the table will be -
1 235.2 \(J\)
2 117.6 \(J\)
3 196 \(J\)
4 98 \(J\)
Explanation:
Mass of the chain hanging \(=4 \times 3=12 \mathrm{~kg}\) Shift in centre of gravity \(=4 / 2=2 m\) \(W=m g h=12 \times 9.8 \times 2=235.2 J\)
PHXI06:WORK ENERGY AND POWER
355849
Force acting on a particle moving in a straight line varies with the velocity of the particle as \(F=K v^{-2}\), where \(K\) is constant. The work done by this force in time \(t\) is:
1 \(K v t\)
2 \(K^{2} v^{2} t^{2}\)
3 \(K^{2} v t\)
4 \(\dfrac{3 K t}{2 v}\)
Explanation:
\(m \dfrac{d v}{d t}=K v^{-2} \int m\left(v^{2} d v\right)=\int K d t\) \(m\left(\dfrac{v^{3}}{3}\right)=K t \quad \dfrac{1}{2} m v^{2}=\dfrac{3}{2} \dfrac{K t}{v}\)
PHXI06:WORK ENERGY AND POWER
355850
A chain of mass \(M\) is placed on a smooth table with \(1 / n\) of its length \(L\) hanging over the edge. The work done in pulling the hanging portion of the chain back to the surface of the table is
1 \(Mg\,L{\rm{/}}n\)
2 \(Mg\,L{\rm{/}}2\,n\)
3 \(Mg\,L{\rm{/}}{n^2}\)
4 \(Mg\,L{\rm{/}}2\,{n^2}\)
Explanation:
\(W = \) change in \(PE\) of \(COM\) of hanging part \( = \frac{M}{n}g\frac{L}{{2n}} = \frac{{MgL}}{{2{n^2}}}\)
355847
A particle is moved from \((0,0)\) to \((2\;m,2\;m)\) with force \(\vec{F}=(x \hat{j}) N\) along path (i) \(y = x\) (ii) \(y=\sqrt{x}\). The work done about two paths is:
1 \(2\;J,2\;J\)
2 \(2J,\frac{8}{3}J\)
3 \(\frac{8}{3}J,\frac{8}{3}J\)
4 \(9\;J,8\;J\)
Explanation:
\(d W=\vec{F} \cdot d \vec{s}=(x \hat{j}) \cdot(d x \hat{i}+d y \hat{j})=x d y\) (i) when \(y=x\) \(W=\int d W=\int_{0}^{2} y d y=\left|\dfrac{y^{2}}{2}\right|_{0}^{2}=2 J\) (ii) When \(y=\sqrt{x}\) or \(y^{2}=x\) \(W=\int_{0}^{2} y^{2} d y=\left|\dfrac{y^{3}}{3}\right|_{0}^{2}=\dfrac{8}{3} J\)
PHXI06:WORK ENERGY AND POWER
355848
A chain of linear mass density 3 \(kg/m\) and length \(8 m\) is lying on the table with 4\(m\) of chain hanging from the edge. The work done in lifting the chain on the table will be -
1 235.2 \(J\)
2 117.6 \(J\)
3 196 \(J\)
4 98 \(J\)
Explanation:
Mass of the chain hanging \(=4 \times 3=12 \mathrm{~kg}\) Shift in centre of gravity \(=4 / 2=2 m\) \(W=m g h=12 \times 9.8 \times 2=235.2 J\)
PHXI06:WORK ENERGY AND POWER
355849
Force acting on a particle moving in a straight line varies with the velocity of the particle as \(F=K v^{-2}\), where \(K\) is constant. The work done by this force in time \(t\) is:
1 \(K v t\)
2 \(K^{2} v^{2} t^{2}\)
3 \(K^{2} v t\)
4 \(\dfrac{3 K t}{2 v}\)
Explanation:
\(m \dfrac{d v}{d t}=K v^{-2} \int m\left(v^{2} d v\right)=\int K d t\) \(m\left(\dfrac{v^{3}}{3}\right)=K t \quad \dfrac{1}{2} m v^{2}=\dfrac{3}{2} \dfrac{K t}{v}\)
PHXI06:WORK ENERGY AND POWER
355850
A chain of mass \(M\) is placed on a smooth table with \(1 / n\) of its length \(L\) hanging over the edge. The work done in pulling the hanging portion of the chain back to the surface of the table is
1 \(Mg\,L{\rm{/}}n\)
2 \(Mg\,L{\rm{/}}2\,n\)
3 \(Mg\,L{\rm{/}}{n^2}\)
4 \(Mg\,L{\rm{/}}2\,{n^2}\)
Explanation:
\(W = \) change in \(PE\) of \(COM\) of hanging part \( = \frac{M}{n}g\frac{L}{{2n}} = \frac{{MgL}}{{2{n^2}}}\)
355847
A particle is moved from \((0,0)\) to \((2\;m,2\;m)\) with force \(\vec{F}=(x \hat{j}) N\) along path (i) \(y = x\) (ii) \(y=\sqrt{x}\). The work done about two paths is:
1 \(2\;J,2\;J\)
2 \(2J,\frac{8}{3}J\)
3 \(\frac{8}{3}J,\frac{8}{3}J\)
4 \(9\;J,8\;J\)
Explanation:
\(d W=\vec{F} \cdot d \vec{s}=(x \hat{j}) \cdot(d x \hat{i}+d y \hat{j})=x d y\) (i) when \(y=x\) \(W=\int d W=\int_{0}^{2} y d y=\left|\dfrac{y^{2}}{2}\right|_{0}^{2}=2 J\) (ii) When \(y=\sqrt{x}\) or \(y^{2}=x\) \(W=\int_{0}^{2} y^{2} d y=\left|\dfrac{y^{3}}{3}\right|_{0}^{2}=\dfrac{8}{3} J\)
PHXI06:WORK ENERGY AND POWER
355848
A chain of linear mass density 3 \(kg/m\) and length \(8 m\) is lying on the table with 4\(m\) of chain hanging from the edge. The work done in lifting the chain on the table will be -
1 235.2 \(J\)
2 117.6 \(J\)
3 196 \(J\)
4 98 \(J\)
Explanation:
Mass of the chain hanging \(=4 \times 3=12 \mathrm{~kg}\) Shift in centre of gravity \(=4 / 2=2 m\) \(W=m g h=12 \times 9.8 \times 2=235.2 J\)
PHXI06:WORK ENERGY AND POWER
355849
Force acting on a particle moving in a straight line varies with the velocity of the particle as \(F=K v^{-2}\), where \(K\) is constant. The work done by this force in time \(t\) is:
1 \(K v t\)
2 \(K^{2} v^{2} t^{2}\)
3 \(K^{2} v t\)
4 \(\dfrac{3 K t}{2 v}\)
Explanation:
\(m \dfrac{d v}{d t}=K v^{-2} \int m\left(v^{2} d v\right)=\int K d t\) \(m\left(\dfrac{v^{3}}{3}\right)=K t \quad \dfrac{1}{2} m v^{2}=\dfrac{3}{2} \dfrac{K t}{v}\)
PHXI06:WORK ENERGY AND POWER
355850
A chain of mass \(M\) is placed on a smooth table with \(1 / n\) of its length \(L\) hanging over the edge. The work done in pulling the hanging portion of the chain back to the surface of the table is
1 \(Mg\,L{\rm{/}}n\)
2 \(Mg\,L{\rm{/}}2\,n\)
3 \(Mg\,L{\rm{/}}{n^2}\)
4 \(Mg\,L{\rm{/}}2\,{n^2}\)
Explanation:
\(W = \) change in \(PE\) of \(COM\) of hanging part \( = \frac{M}{n}g\frac{L}{{2n}} = \frac{{MgL}}{{2{n^2}}}\)
NEET Test Series from KOTA - 10 Papers In MS WORD
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PHXI06:WORK ENERGY AND POWER
355847
A particle is moved from \((0,0)\) to \((2\;m,2\;m)\) with force \(\vec{F}=(x \hat{j}) N\) along path (i) \(y = x\) (ii) \(y=\sqrt{x}\). The work done about two paths is:
1 \(2\;J,2\;J\)
2 \(2J,\frac{8}{3}J\)
3 \(\frac{8}{3}J,\frac{8}{3}J\)
4 \(9\;J,8\;J\)
Explanation:
\(d W=\vec{F} \cdot d \vec{s}=(x \hat{j}) \cdot(d x \hat{i}+d y \hat{j})=x d y\) (i) when \(y=x\) \(W=\int d W=\int_{0}^{2} y d y=\left|\dfrac{y^{2}}{2}\right|_{0}^{2}=2 J\) (ii) When \(y=\sqrt{x}\) or \(y^{2}=x\) \(W=\int_{0}^{2} y^{2} d y=\left|\dfrac{y^{3}}{3}\right|_{0}^{2}=\dfrac{8}{3} J\)
PHXI06:WORK ENERGY AND POWER
355848
A chain of linear mass density 3 \(kg/m\) and length \(8 m\) is lying on the table with 4\(m\) of chain hanging from the edge. The work done in lifting the chain on the table will be -
1 235.2 \(J\)
2 117.6 \(J\)
3 196 \(J\)
4 98 \(J\)
Explanation:
Mass of the chain hanging \(=4 \times 3=12 \mathrm{~kg}\) Shift in centre of gravity \(=4 / 2=2 m\) \(W=m g h=12 \times 9.8 \times 2=235.2 J\)
PHXI06:WORK ENERGY AND POWER
355849
Force acting on a particle moving in a straight line varies with the velocity of the particle as \(F=K v^{-2}\), where \(K\) is constant. The work done by this force in time \(t\) is:
1 \(K v t\)
2 \(K^{2} v^{2} t^{2}\)
3 \(K^{2} v t\)
4 \(\dfrac{3 K t}{2 v}\)
Explanation:
\(m \dfrac{d v}{d t}=K v^{-2} \int m\left(v^{2} d v\right)=\int K d t\) \(m\left(\dfrac{v^{3}}{3}\right)=K t \quad \dfrac{1}{2} m v^{2}=\dfrac{3}{2} \dfrac{K t}{v}\)
PHXI06:WORK ENERGY AND POWER
355850
A chain of mass \(M\) is placed on a smooth table with \(1 / n\) of its length \(L\) hanging over the edge. The work done in pulling the hanging portion of the chain back to the surface of the table is
1 \(Mg\,L{\rm{/}}n\)
2 \(Mg\,L{\rm{/}}2\,n\)
3 \(Mg\,L{\rm{/}}{n^2}\)
4 \(Mg\,L{\rm{/}}2\,{n^2}\)
Explanation:
\(W = \) change in \(PE\) of \(COM\) of hanging part \( = \frac{M}{n}g\frac{L}{{2n}} = \frac{{MgL}}{{2{n^2}}}\)
