Newton’s Second Law for Rotation Examples 1. A cylinder having a mass of 2.0 kg can rotate about an axis through its center O. Forces are applied as in the diagram: F1 = 6.0 N, F2 = 4.0 N, F3 = 2.0 N, F4 = 5.0 N. Also, R1 = 5.0 cm and R2 = 12 cm. Calculate the magnitude and direction of the angular acceleration of the cylinder, assuming that during rotation, the forces maintain their same angles relative to the cylinder. But

Newton’s Second Law for Rotation Examples 1. A cylinder having a mass of 2.0 kg can rotate about an axis through its center O. Forces are applied as in the diagram: F1 = 6.0 N, F2 = 4.0 N, F3 = 2.0 N, F4 = 5.0 N. Also, R1 = 5.0 cm and R2 = 12 cm. Calculate the magnitude and direction of the angular acceleration of the cylinder, assuming that during rotation, the forces maintain their same angles relative to the cylinder.

Newton’s Second Law for Rotation Examples 1. A cylinder having a mass of 2.0 kg can rotate about an axis through its center O. Forces are applied as in the diagram: F1 = 6.0 N, F2 = 4.0 N, F3 = 2.0 N, F4 = 5.0 N. Also, R1 = 5.0 cm and R2 = 12 cm. Calculate the magnitude and direction of the angular acceleration of the cylinder, assuming that during rotation, the forces maintain their same angles relative to the cylinder. Units:

Newton’s Second Law for Rotation Examples 2. The massive shield door at the Lawrence Livermore Laboratory is the world’s heaviest hinged door. The door has a mass of 44 000 kg, a rotational inertia about an axis through its hinges of 8.7 x 104 kgm2, and a width of 2.4 m. Neglecting friction, what steady force, applied at its outer edge and perpendicular to the plane of the door, can move it from rest through an angle of 90.° in 30. s?

Newton’s Second Law for Rotation Examples 2. The massive shield door at the Lawrence Livermore Laboratory is the world’s heaviest hinged door. The door has a mass of 44 000 kg, a rotational inertia about an axis through its hinges of 8.7 x 104 kgm2, and a width of 2.4 m. Neglecting friction, what steady force, applied at its outer edge and perpendicular to the plane of the door, can move it from rest through an angle of 90.° in 30. s?

Newton’s Second Law for Rotation Examples 2. The massive shield door at the Lawrence Livermore Laboratory is the world’s heaviest hinged door. The door has a mass of 44 000 kg, a rotational inertia about an axis through its hinges of 8.7 x 104 kgm2, and a width of 2.4 m. Neglecting friction, what steady force, applied at its outer edge and perpendicular to the plane of the door, can move it from rest through an angle of 90.° in 30. s?

Newton’s Second Law for Rotation Examples 2. The massive shield door at the Lawrence Livermore Laboratory is the world’s heaviest hinged door. The door has a mass of 44 000 kg, a rotational inertia about an axis through its hinges of 8.7 x 104 kgm2, and a width of 2.4 m. Neglecting friction, what steady force, applied at its outer edge and perpendicular to the plane of the door, can move it from rest through an angle of 90.° in 30. s?

Newton’s Second Law for Rotation Examples 3. Two identical blocks, each of mass M, are connected by a massless string over a disk pulley of radius R, and mass M. The string does not slip on the pulley, it is not known whether or not there is friction between the table and the sliding block; the pulley’s axis is frictionless. When this system is released it is found that the pulley turns through an angle θ in a time t and the acceleration of the blocks is constant. Express all answers in terms of M, R, θ, t, and g. a. What is the angular acceleration of the pulley? Acceleration is uniform

Newton’s Second Law for Rotation Examples 3. Two identical blocks, each of mass M, are connected by a massless string over a disk pulley of radius R, and mass M. The string does not slip on the pulley, it is not known whether or not there is friction between the table and the sliding block; the pulley’s axis is frictionless. When this system is released it is found that the pulley turns through an angle θ in a time t and the acceleration of the blocks is constant. Express all answers in terms of M, R, θ, t, and g. b. What is the linear acceleration of the two blocks? Since the string does not slip on the pulley, the linear acceleration of the outside of the pulley must be the same as the linear accelerations of the blocks.

Newton’s Second Law for Rotation Examples 3. Two identical blocks, each of mass M, are connected by a massless string over a disk pulley of radius R, and mass M. The string does not slip on the pulley, it is not known whether or not there is friction between the table and the sliding block; the pulley’s axis is frictionless. When this system is released it is found that the pulley turns through an angle θ in a time t and the acceleration of the blocks is constant. Express all answers in terms of M, R, θ, t, and g. c. What are the tensions in the upper (T1) and lower (T2) sections of the string? Mass on Table T1 and Ffk are unknown.

Newton’s Second Law for Rotation Examples 3. Two identical blocks, each of mass M, are connected by a massless string over a disk pulley of radius R, and mass M. The string does not slip on the pulley, it is not known whether or not there is friction between the table and the sliding block; the pulley’s axis is frictionless. When this system is released it is found that the pulley turns through an angle θ in a time t and the acceleration of the blocks is constant. Express all answers in terms of M, R, θ, t, and g. c. What are the tensions in the upper (T1) and lower (T2) sections of the string? Hanging Mass

Newton’s Second Law for Rotation Examples 3. Two identical blocks, each of mass M, are connected by a massless string over a disk pulley of radius R, and mass M. The string does not slip on the pulley, it is not known whether or not there is friction between the table and the sliding block; the pulley’s axis is frictionless. When this system is released it is found that the pulley turns through an angle θ in a time t and the acceleration of the blocks is constant. Express all answers in terms of M, R, θ, t, and g. c. What are the tensions in the upper (T1) and lower (T2) sections of the string? Notice that T2 < Fg as required

Newton’s Second Law for Rotation Examples 3. Two identical blocks, each of mass M, are connected by a massless string over a disk pulley of radius R, and mass M. The string does not slip on the pulley, it is not known whether or not there is friction between the table and the sliding block; the pulley’s axis is frictionless. When this system is released it is found that the pulley turns through an angle θ in a time t and the acceleration of the blocks is constant. Express all answers in terms of M, R, θ, t, and g. c. What are the tensions in the upper (T1) and lower (T2) sections of the string? Disk pulley Since α is clockwise

Newton’s Second Law for Rotation Examples 3. Two identical blocks, each of mass M, are connected by a massless string over a disk pulley of radius R, and mass M. The string does not slip on the pulley, it is not known whether or not there is friction between the table and the sliding block; the pulley’s axis is frictionless. When this system is released it is found that the pulley turns through an angle θ in a time t and the acceleration of the blocks is constant. Express all answers in terms of M, R, θ, t, and g. c. What are the tensions in the upper (T1) and lower (T2) sections of the string? But T1 and T2 are tangent to the pulley so θ = 90.º and sin 90.º = 1

Newton’s Second Law for Rotation Examples 3. Two identical blocks, each of mass M, are connected by a massless string over a disk pulley of radius R, and mass M. The string does not slip on the pulley, it is not known whether or not there is friction between the table and the sliding block; the pulley’s axis is frictionless. When this system is released it is found that the pulley turns through an angle θ in a time t and the acceleration of the blocks is constant. Express all answers in terms of M, R, θ, t, and g. c. What are the tensions in the upper (T1) and lower (T2) sections of the string? Notice that T2 > T1 as required