How long does it take for a cassette tape to unwind?

00:31 Hint given | 01:00 Solution starts | 02:32 Answer

Question GM1L2: Concepts used in solution

Kinematics

Problem statement

While listening to a cassette, it took time t for the radius of the unwinding tape reel to be reduced by half. How long will it take for the radius to be reduced by a factor of two again?

Instantaneous axis of rotation and other ways to find velocity of points on a moving triangle.

00:29 Hints given | 02:54 First solution | 06:31 Second solution | 08:40 Third solution | 12:07 Answer

Question R4L2: Concepts used in solution

Instantaneous axis of rotation | Relative velocity |  Velocity due to rotation

Instantaneous axis of rotation explained in MIT OCW

Definition and simulation in 3D available here

Problem statement

A, B, and C are vertices of a moving rigid equilateral triangle. At a particular instant, A moves with velocity u along AB towards B. C moves away from B along BC with an unknown speed. Find the velocity of B.

How long does a rotating square plate take to stop?

00:36 Hint given | 02:28 Solution starts | 06:47 Answer

Question R2L2: Concepts used in solution

Moment of inertia | Dimension analysis | Torque | Angular impulse

Problem statement

A uniform square plate is placed on a rough horizontal floor hinged at one corner. When it is given an angular velocity w about a vertical axis through this corner, as shown in figure(a), it takes time t to come to a complete stop. Now the same square plate is hinged about the center and given the same angular velocity about a vertical axis as shown in figure(b). How long will it take to come to a complete stop now?

A fine way to find the moment of inertia of a thin stick without using calculus

00:39 Hint given | 01:26 Solution starts | 05:29 Answer

Question R1L2: Concepts used in solution

Moment of inertia | Dimension analysis | Parallel axis theorem | Scaling

Derivation of parallel axis theorem from MIT OCW

Problem statement

Using just parallel axis theorem and scaling argument, find the moment of inertia of a thin stick about an axis passing through its center and perpendicular to its length. On what other shapes can this idea be used?

Finding the gravitational field inside a cavity in a solid sphere

00:39 Hint given | 01:26 Solution starts | 05:29 Answer

Question G1L2: Concepts used in solution

Shell theorem | Superposition principle | Vectors

Shell theorem at Hyperphysics and Wikipedia

Problem statement

A sphere of radius R and uniform density p has within it a spherical cavity of radius r whose center is as shown in the figure. Find the magnitude and direction of the gravitational field at a point r within the cavity. What is the direction of the initial motion of a point mass placed somewhere inside the cavity?

Could a pure rolling disc have zero angular momentum?

00:41 Hint given | 01:50 Solution starts | 07:20 Answer

Question R3L2: Concepts used in solution

L due to translation and rotation | Vector cross product | Right hand rule for L |

Derivation of L formula shared by MIT can be found here

MIT OCW video on Angular Momentum of Translation and Rotation

Problem statement

A disc of radius R and mass M rolls without slipping in the direction of the positive x-axis with speed u. Are there points about which the angular momentum of the disc is zero?

How do the forces on a slowly pulled ball change as it moves over a hemisphere?

01:00 Solution starts | 02:33 Hint given | 04:17 Answer

Question N1L2: Concepts used in solution

Equilibrium | Vector addition | Properties of triangles

Problem statement

A small metal ball is being pulled slowly from the bottom to the top of a fixed frictionless hemisphere, as shown in the figure. Assuming that the radii of the ball and the pulley are much smaller than that of the hemisphere, do the magnitudes of the pulling force F and the contact force N between the ball and the hemisphere increase, decrease or remain unchanged?

What is the maximum angle at which a ball thrown keeps going away from you?

00:54 Hint given | 04:35 Solution starts | 08:51 Answer | 09:13 Animation

Animate projectile motion and watch the distance from origin vary. Change angle or speed and see the effect.

Animated on Desmos

Question P1L2: Concepts used in solution

Equations of motion | Vectors | Vector dot product

Click here to play and explore quadratic equations  and the role of the discriminant

Problem statement

What is the maximum angle at which a projectile can be thrown such that the distance from the launch point keeps increasing with time?

How to replace a pulley and two masses with an effective mass

00:40 Hint given | 01:46 Solution starts | 04:34 M from limits | 11:13 M from F=ma

Question N2L2: Concepts used in solution

Limits |  F=ma | Simple two mass pulley system

Problem statement

A ball of mass m is connected by rope to a pulley (P) on the other side as shown in figure (a). Balls of mass m1 and m2 are tied to the ends of a rope that passes over this pulley (P). Find the acceleration of mass m, assuming the pulleys and ropes to be massless and frictionless. Find the equivalent mass M of a new ball that could replace the pulley (P) and masses m1 and m2, as shown in figure (b), but provide the correct acceleration for mass m.