The figure above shows a small sphere of mass m at a height H from the center of a uniform ring of radius R and mass M. The center of the ring is placed at the origin of a Cartesian coordinate system. The x and y directions line in the plane of the ring, while the z-direction is positive upwards. Part (a) Take the x-axis to be directed towards the right (in the plane of the ring). Write an expression for the x-component of the gravitational force Fx on the mass m, where G is the gravitational constant. Part (b) Take the y-axis to be directed out of the board (in the plane of the ring). Write an expression for the y-component of the gravitational force Fy on the mass m, where G is the gravitational constant. Part (c) Write an expression for the z-component of the gravitational force Fz on the mass m, where G is the gravitational constant. Part (d) The ring has a mass of 7.5 kg and the small sphere's mass is half that of the ring. The height is H = 0.75 m and the ring has a radius of 0.75 m. What is the magnitude of the gravitational force, in newtons, of the ball?
Rigid Body
A rigid body is an object which does not change its shape or undergo any significant deformation due to an external force or movement. Mathematically speaking, the distance between any two points inside the body doesn't change in any situation.
Rigid Body Dynamics
Rigid bodies are defined as inelastic shapes with negligible deformation, giving them an unchanging center of mass. It is also generally assumed that the mass of a rigid body is uniformly distributed. This property of rigid bodies comes in handy when we deal with concepts like momentum, angular momentum, force and torque. The study of these properties – viz., force, torque, momentum, and angular momentum – of a rigid body, is collectively known as rigid body dynamics (RBD).
The figure above shows a small sphere of mass m at a height H from the center of a uniform ring of radius R and mass M. The center of the ring is placed at the origin of a Cartesian coordinate system. The x and y directions line in the plane of the ring, while the z-direction is positive upwards.
Part (a) Take the x-axis to be directed towards the right (in the plane of the ring). Write an expression for the x-component of the gravitational force Fx on the mass m, where G is the gravitational constant.
Part (b) Take the y-axis to be directed out of the board (in the plane of the ring). Write an expression for the y-component of the gravitational force Fy on the mass m, where G is the gravitational constant.
Part (c) Write an expression for the z-component of the gravitational force Fz on the mass m, where G is the gravitational constant.
Part (d) The ring has a mass of 7.5 kg and the small sphere's mass is half that of the ring. The height is H = 0.75 m and the ring has a radius of 0.75 m. What is the magnitude of the gravitational force, in newtons, of the ball?
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