A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. (a) Will it reach the bottom with the same speed in each case? (b) Will it take longer to roll down one plane than the other? (c) If so, which one and why?
(a)
The equation for the moment of inertia of a solid sphere is,
I = 2 5 m r 2
Here, m is the mass and r is the radius of the solid sphere.
The equation for the velocity of the center of mass of the solid sphere is,
v = r ω
Here, ω 0 is the angular speed of the solid sphere.
The equation for the initial potential energy of the solid sphere is,
U 0 = m g h
Here, g is the gravitational acceleration and h is the vertical distance covered by the solid sphere.
As the solid sphere is released from rest, the initial kinetic energy of the solid sphere is zero.
K 0 = 0
The equation for the final kinetic energy of the solid sphere is,
K f = 1 2 m v 2 + 1 2 I ω 2
Substitute the values of I and ω in the above equation.
K f = 1 2 m v 2 + 1 2 ( 2 5 m r 2 ) ω 2 = 1 2 m v 2 + 1 5 m ( r ω ) 2 = 1 2 m v 2 + 1 5 m ( v ) 2 = 7 10 m v 2
As the solid sphere finally reaches the bottom of the plane, the final potential energy of the solid sphere is zero, i.e,
U f = 0
The expression for the law of conservation of energy is,
U 0 + K 0 = U f + K f
Substitute the values of U 0 , K 0 , U f ,and K f in the above equation.
m g h + 0 = 0 + 7 10 m v 2 v 2 = ( 10 7 ) g h v = ( 10 7 ) g h
The final speed only depends on the gravitational acceleration g and vertical height h . The angle of inclination does not affect the speed of the solid sphere at the bottom of the plane. Therefore, the speed of the solid sphere at the bottom of the plane will remain same at any angle of inclination.
Thus, the solid sphere will reach the bottom with the same speed in each case.
(b)
The figure of motion of the solid sphere is shown below:
The kinematic equation for motion along the inclined plane is,
v = u + ( g sin θ ) t …… (1)
Here, v is the final velocity of the sphere, θ is the angle of inclination, and t is the time duration of motion.
From the above equation,
t ∝ 1 sin θ
The value of sin θ increases as θ increases, therefore the time duration of motion decreases as the angle of inclination increases.
Thus, in case of smaller inclination angle, the solid sphere will take longer time to roll down the plane.
(c)
From the equation (1),
t ∝ 1 sin θ
And,
sin θ ∝ θ
Therefore,
t ∝ 1 θ
Thus, the plane with smaller angle of inclination will take longer time duration to reach the bottom of the plane.
(a)
The equation for the moment of inertia of a solid sphere is,
Here,
The equation for the velocity of the center of mass of the solid sphere is,
Here,
The equation for the initial potential energy of the solid sphere is,
Here,
As the solid sphere is released from rest, the initial kinetic energy of the solid sphere is zero.
The equation for the final kinetic energy of the solid sphere is,
Substitute the values of
As the solid sphere finally reaches the bottom of the plane, the final potential energy of the solid sphere is zero, i.e,
The expression for the law of conservation of energy is,
Substitute the values of
The final speed only depends on the gravitational acceleration
Thus, the solid sphere will reach the bottom with the same speed in each case.
(b)
The figure of motion of the solid sphere is shown below:
The kinematic equation for motion along the inclined plane is,
Here,
From the above equation,
The value of
Thus, in case of smaller inclination angle, the solid sphere will take longer time to roll down the plane.
(c)
From the equation (1),
And,
Therefore,
Thus, the plane with smaller angle of inclination will take longer time duration to reach the bottom of the plane.
