Asolid sphere rolls down two different inclined planes of the sameheights but different angles of inclination. (a) Will it reach thebottom with the same speed in each case? (b) Will it take longer toroll down one plane than the other? (c) If so, which one and why?
Asolid sphere rolls down two different inclined planes of the sameheights but different angles of inclination. (a) Will it reach thebottom with the same speed in each case? (b) Will it take longer toroll down one plane than the other? (c) If so, which one and why?
Answer:(a)Yes (b)Yes (c) Onthe smaller inclination
(a)Massof the sphere = m
Heightof the plane = h
Velocityof the sphere at the bottom of the plane =v
Atthe top of the plane, the total energy of the sphere = Potentialenergy = mgh
Atthe bottom of the plane, the sphere hasboth translational and rotational kinetic energies.
Hence,total energy = 
Usingthe law of conservation of energy, we can write:
Fora solid sphere, the moment of inertia about its centre, 
Hence,equation (i)becomes:
Hence,the velocity of the sphere at the bottom depends only on height (h)and acceleration due to gravity (g). Both these values are constants.Therefore, the velocity at the bottom remains the same from whicheverinclined plane the sphere is rolled.
(b),(c)Considertwo inclined planes with inclinations θ1and θ2,related as:
θ1< θ2
Theacceleration produced in the sphere when it rolls down the planeinclined at θ1is:
gsin θ1
The various forces acting on the sphere are shown in the followingfigure.
R1is the normal reaction to the sphere.
Similarly,the acceleration produced in the sphere when it rolls down the planeinclined at θ2is:
gsin θ2
Thevarious forces acting on the sphere areshown in the following figure.
R2is the normal reaction to the sphere.
θ2> θ1;sin θ2> sin θ1 ... (i)
∴a2> a1 … (ii)
Initialvelocity, u =0
Finalvelocity, v= Constant
Usingthe first equation of motion, we can obtainthe time of roll as:
v= u + at
Fromequations (ii)and (iii),we get:
t2< t1
Hence, the sphere will take a longer time to reach the bottom of theinclined plane having the smaller inclination.
Answer:(a)Yes (b)Yes (c) Onthe smaller inclination
(a)Massof the sphere = m
Heightof the plane = h
Velocityof the sphere at the bottom of the plane =v
Atthe top of the plane, the total energy of the sphere = Potentialenergy = mgh
Atthe bottom of the plane, the sphere hasboth translational and rotational kinetic energies.
Hence,total energy =
Usingthe law of conservation of energy, we can write:
Fora solid sphere, the moment of inertia about its centre,
Hence,equation (i)becomes:
Hence,the velocity of the sphere at the bottom depends only on height (h)and acceleration due to gravity (g). Both these values are constants.Therefore, the velocity at the bottom remains the same from whicheverinclined plane the sphere is rolled.
(b),(c)Considertwo inclined planes with inclinations θ1and θ2,related as:
θ1< θ2
Theacceleration produced in the sphere when it rolls down the planeinclined at θ1is:
gsin θ1
The various forces acting on the sphere are shown in the followingfigure.
R1is the normal reaction to the sphere.
Similarly,the acceleration produced in the sphere when it rolls down the planeinclined at θ2is:
gsin θ2
Thevarious forces acting on the sphere areshown in the following figure.
R2is the normal reaction to the sphere.
θ2> θ1;sin θ2> sin θ1 ... (i)
∴a2> a1 … (ii)
Initialvelocity, u =0
Finalvelocity, v= Constant
Usingthe first equation of motion, we can obtainthe time of roll as:
v= u + at
Fromequations (ii)and (iii),we get:
t2< t1
Hence, the sphere will take a longer time to reach the bottom of theinclined plane having the smaller inclination.
