Question

A solid sphere rolling without friction on a horizontal surface with a constant speed of \(2 \mathrm{~m} / \mathrm{s}\), rolls up on an inclined ramp which is inclined at \(30^{\circ}\). The maximum distance travelled by the sphere on the inclined ramp is (acceleration due to gravity \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2, \sin 30^{\circ}=\frac{1}{2}\) )

• A. 56 cm
• B. 25 cm
• C. 47 cm
• D. 30 cm

Hint:

Use energy conservation including both translational and rotational kinetic energy of the solid sphere.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
A solid sphere rolling without slipping has total kinetic energy \(\dfrac12mv^2\left(1+\dfrac25\right)=\dfrac{7}{10}mv^2\).

Step 2: Energy conservation
\(\dfrac{7}{10}mv^2=mgh\Rightarrow h=\dfrac{7v^2}{10g}=\dfrac{7\times4}{100}=0.28\text{ m}\).

Step 3: Distance along the ramp
\(s=\dfrac{h}{\sin30^{\circ}}=\dfrac{0.28}{1/2}=0.56\text{ m}=56\text{ cm}\).

Final Answer:
The sphere travels 56 cm up the ramp. \[ \boxed{\text{(A) }56\text{ cm}} \]