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Byju's Answer
Standard XII
Physics
Race of Rollers
A solid spher...
Question
A solid sphere and a solid cylinder having the same mass and radius, roll down the same incline. Find the ratio of their acceleration.
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Solution
Given: A solid sphere and a solid cylinder having the same mass and radius, roll down the same incline.
To find the ratio of their acceleration.
Solution:
From the above fig:
m
g
sin
θ
−
f
=
m
a
,
where
f
is the frictional force
Now for solid cylinder,
f
1
R
=
I
α
⟹
f
1
R
=
m
R
2
2
×
a
1
R
⟹
f
1
=
m
a
1
2
∴
m
g
sin
θ
=
m
a
1
+
f
1
⟹
m
g
sin
θ
=
m
a
+
m
a
1
2
=
3
2
m
a
1
⟹
a
1
=
2
3
sin
θ
.
.
.
.
.
.
(
i
)
Now for solid sphere,
f
2
R
=
I
α
⟹
f
2
R
=
2
m
R
2
5
×
a
2
R
⟹
f
2
=
2
m
a
2
5
∴
m
g
sin
θ
=
m
a
2
+
f
2
⟹
m
g
sin
θ
=
m
a
2
+
2
m
a
2
5
=
7
2
m
a
2
⟹
a
2
=
2
7
sin
θ
.
.
.
.
.
.
(
i
)
Hence the ratio of their acceleration is:
a
2
:
a
1
=
2
sin
θ
7
:
2
sin
θ
3
=
3
:
7
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