A solid sphere of mass *M* and radius *R* rolls without slipping down a ramp inclined at an angle *θ* to the horizontal. The moment of inertia of a solid sphere about its centre is *I* = (2/5)*MR*2.
(a) (i) On the diagram, draw and label the forces acting on the sphere.
[2]
(ii) State the condition required for the sphere to roll without slipping in terms of its linear velocity *v* and angular velocity *ω*.
[1]
(b) Show that the linear acceleration *a* of the centre of mass of the sphere down the ramp is given by *a* = (5/7)*g*sin*θ*.
[4]
(c) The sphere starts from rest and rolls a distance *s* along the ramp. Determine, in terms of *M*, *g*, *s* and *θ*, the total kinetic energy of the sphere after rolling distance *s*.
[3]
Question
(i) On the diagram, draw and label the forces acting on the sphere.
[2]
Hint
Hint
Hint
Identify all forces acting on the sphere. Consider gravity, the contact force from the ramp (which has two components: normal and friction), and ensure forces are drawn from the correct points of application.
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
Forces: Weight (Mg) acting vertically downwards from the centre of mass. Normal force (N) acting perpendicular to the ramp surface, outwards from the point of contact. Static friction force (f) acting parallel to the ramp surface, upwards along the ramp from the point of contact.
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
(ii) State the condition required for the sphere to roll without slipping in terms of its linear velocity *v* and angular velocity *ω*.
[1]
Hint
Hint
Hint
Rolling without slipping means the point of contact is instantaneously at rest relative to the surface. How does this relate the linear speed of the centre of mass to the angular speed?
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
\\(v = \\omega R\\)
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
Show that the linear acceleration *a* of the centre of mass of the sphere down the ramp is given by *a* = (5/7)*g*sin*θ*.
[4]
Hint
Hint
Hint
Apply Newton’s second law for linear motion down the ramp (consider forces parallel to the ramp). Apply Newton’s second law for rotation about the centre of mass (consider torques). Use the no-slip condition \\(a = \\alpha R\\) to relate linear and angular acceleration.
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
Derivation: Apply \\(F_{net, parallel} = Ma\\) and \\(\\tau_{net} = I\\alpha\\). Substitute \\(f\\) from the torque equation into the force equation, using \\(I = \\frac{2}{5}MR^2\\) and \\(\\alpha = a/R\\). Solve for \\(a\\) to get \\(a = \\frac{5}{7}g\\sin\\theta\\).
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
The sphere starts from rest and rolls a distance *s* along the ramp. Determine, in terms of *M*, *g*, *s* and *θ*, the total kinetic energy of the sphere after rolling distance *s*. [3]
Hint
Hint
Hint
Use the work-energy theorem. The net work done on the sphere equals its change in kinetic energy. The work done by gravity is related to the vertical height descended. Alternatively, find the final velocity using kinematics and then calculate both translational and rotational kinetic energy.
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
Total KE = \\(Mgs\\sin\\theta\\)
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
