Gravitational potential energy formula for a rigid body + angular velocity of a falling rod.
00:00 Introduction: we're going to derive a formula for the gravitational potential energy of a rigid body, then apply the formula to an example. We break the rigid body into point masses, m_1 through m_n, and we also give a reminder of the center of mass and center of mass position vector. We break the center of mass position vector into components, and we emphasize that the y coordinate of the center of mass is the important part because gravitational potential energy depends on the y coordinate. 00:14 We set up the sum of the potential energies of all the point masses in the rigid body, and we begin to manipulate the sum. The g is common to each term and we can factor this out. Then we recognize the sum m_i*y_i as part of the center of mass y coordinate formula: it's equal to M*y_cm. So, we arrive at the gravitational potential energy formula for a rigid body: Mgy_cm, and the interpretation of this formula is that we can compute the gravitational potential energy by pretending all the mass in a body is located at the center of mass! 02:32 Example of using the rigid body gravitational potential energy formula: a hinged rod falls from the horizontal to vertical position, and we want to compute angular velocity of a falling rod. We compute the initial and final y coordinates of the center of mass, and write down the energy conservation formula, where the final kinetic energy is pure rotational kinetic energy. We plug in the moment of inertia for a rod rotating about one end, and we solve for the final angular velocity of the rod. Finally, we check the units on our answer to verify that they come out to rad/s and we're done!
00:00 Introduction: we're going to derive a formula for the gravitational potential energy of a rigid body, then apply the formula to an example. We break the rigid body into point masses, m_1 through m_n, and we also give a reminder of the center of mass and center of mass position vector. We break the center of mass position vector into components, and we emphasize that the y coordinate of the center of mass is the important part because gravitational potential energy depends on the y coordinate. 00:14 We set up the sum of the potential energies of all the point masses in the rigid body, and we begin to manipulate the sum. The g is common to each term and we can factor this out. Then we recognize the sum m_i*y_i as part of the center of mass y coordinate formula: it's equal to M*y_cm. So, we arrive at the gravitational potential energy formula for a rigid body: Mgy_cm, and the interpretation of this formula is that we can compute the gravitational potential energy by pretending all the mass in a body is located at the center of mass! 02:32 Example of using the rigid body gravitational potential energy formula: a hinged rod falls from the horizontal to vertical position, and we want to compute angular velocity of a falling rod. We compute the initial and final y coordinates of the center of mass, and write down the energy conservation formula, where the final kinetic energy is pure rotational kinetic energy. We plug in the moment of inertia for a rod rotating about one end, and we solve for the final angular velocity of the rod. Finally, we check the units on our answer to verify that they come out to rad/s and we're done!