A bowling ball of mass M1=6.0 kg is initially at rest on the sloped side of a wedge of mass M2=

step 1 So we can first say that the net force acting on the ball, applying Newton's second law on the b

A bowling ball of mass M1=6.0 kg is initially at rest on...

A bowling ball of mass $M_{1}=6.0 \mathrm{~kg}$ is initially at rest on the sloped side of a wedge of mass $M_{2}=9.0 \mathrm{~kg}$ that is on a frictionless horizontal floor. The side of the wedge is sloped at an angle of $\theta=36.9^{\circ}$ above the horizontal. a) With what magnitude of horizontal force should the wedge be pushed to keep the bowling ball at a constant height on the slope? b) What is the magnitude of the acceleration of the wedge, if no external force is applied?

A bowling ball of mass $M_{1}=6.0 \mathrm{~kg}$ is initially at rest on the sloped side of a wedge of mass $M_{2}=9.0 \mathrm{~kg}$ that is on a frictionless horizontal floor. The side of the wedge is sloped at an angle of $\theta=36.9^{\circ}$ above the horizontal.
a) With what magnitude of horizontal force should the wedge be pushed to keep the bowling ball at a constant height on the slope?
b) What is the magnitude of the acceleration of the wedge, if no external force is applied?

Key Concepts

Constraint Conditions on Relative Motion

This concept involves applying constraints that limit the relative motion of bodies in contact, such as the requirement that the bowling ball remains at a constant height on the inclined plane of the wedge. These conditions lead to relationships between the applied forces and the accelerations of the bodies, thereby enabling the determination of the force required to maintain a specific relative motion or the acceleration when no external force is applied.

Force Decomposition

Force decomposition involves breaking forces into components along chosen directions, typically parallel and perpendicular to a particular surface or axis. In the context of a wedge inclined at an angle, decomposing forces such as gravity and inertial (pseudo) forces into components along and perpendicular to the plane allows for the correct application of Newton's laws to ensure no net acceleration in the vertical (or normal) direction.

Newton's Second Law

This concept states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). In problems involving accelerating systems—as with the wedge and bowling ball—the application of Newton's second law to each body separately is vital for developing the equations of motion and understanding how forces influence acceleration.

Non-Inertial Reference Frames and Pseudo Forces

When analyzing motion from the perspective of an accelerating frame (like that of the moving wedge), fictitious or pseudo forces must be introduced to account for the apparent forces felt by objects within the accelerating frame. This concept is crucial for correctly setting up the force balance from the viewpoint of the wedge and for enforcing the condition of constant height of the bowling ball along the slope.

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