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A particle of mass m moves down the inner surface of a hemisphere without friction. Find the equations of motion.

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A basket of flowers of mass 3 kg is placed on a flat grassy slope that makes an angle θ with the horizontal. The coefficient of static friction between the basket and the slope is 0.45 and the basket is on the point of slipping down the slope. Model the basket of flowers as a particle and the grassy slope as a plane. Take the magnitude of the acceleration due to gravity, g, to be 9.8 m s−2 . State the forces that act on the basket. Draw a force diagram showing them, defining the symbols that you use to represent the forces and marking the sizes of the angles to show the directions of the forces. Take the unit vector i to point up the slope and the unit vector j to point upwards and perpendicular to the slope. Add these unit vectors to your force diagram.
A block weighing 14.0 N can slide without friction on an inclined plane at an angle of 40.0 degrees. At one end of the beam is attached a light spring which has a length when not stretched of 0.450 m and a spring constant of 120 N / m. determine the equation of motion of the block by taking the xy coordinates (horizontal x and y vertical)!
Two blocks of masses m, = 4.0 kg and mg = 8.0 kg are connected with a string that passes over a very light pulley (Figure 1). Friction in the pulley can be ignored. Block 1 is resting on a rough table and block 2 is hanging over the edge. The coefficient of friction between the block 1 and the table is 0.70 (assume static and kinetic friction have the same value). Block 1 is also connected to a spring with a constant 300 N/m. In the initial state, the spring is relaxed as a person is holding block 2, but the string is still taut. When block 2 is released, it moves down for a distanced until it stops (the final state). Figure 1 of 1
A system is made up of a pulley of radius a with no friction and mass , as well as a massless string of length L that is tied to some point masses m1 hanging on the left and some point masses m2 hanging on the right. Determine the system's acceleration by expressing it with a single degree of freedom.
In the friction experiment, if the friction block has a mass of 599 g, and it started to move at a constant speed after giving it a slight push when the mass in the hanger is 284 g (including the hanger), what is the coefficient of kinetic friction?
Here we have a block of mass (m=2.50kg) resting on a place inclined at an angle of ø=30° to the horizontal. The coefficient of static friction between the block and the plane is Ustatic = 0.135 and the block is stationary but just on the point of sliding up the slope. The diagram shows the four forces acting on the block: an applied force F1 acting up the slope, the block's weight mg, the normal reaction force N and the force of static friction, Ff. In this case, the force of static friction acts down the slope, opposing the tendency of the block to move up the slope. Find the the maximum magnitude of the applied force F1 that can be exerted if the block is to remain stationary.
k A 13.00 kg particle starts from the origin at time zero. Its velocity as a function of time is given by = 10t² + 1tj where is in meters per second and t is in seconds. (Use the following as necessary: t.) V (a) Find its position as a function of time. ŕ= esc 10/³1 + ²/2 2²; 3 (b) Describe its motion qualitatively. The particle starts from rest at the origin, starts moving in the y direction, and gains speed faster and faster while turning to move more and more nearly parallel to the x axis. Score: 0.06 out of 0.06 Comment: (c) Find its acceleration as a function of time. a = 20ti + lj m/s² (d) Find the net force exerted on the particle as a function of time. F= T= (e) Find the net torque about the origin exerted on the particle as a function of time. N m X X (f) Find the angular momentum of the particle as a function of time. Ľ Need Help? Read It F1 N x (g) Find the kinetic energy of the particle as a function of time. K= x kg m²/s X (h) Find the power injected into the particle as a…
A block of mass m rests on a plane inclined an angle with the horizontal. A spring with force constant k is attached to the block. The coefficient of static friction between the block and plane is  s. The spring is pulled upward along the plane very slowly. (a) What is the extension of the spring the instant the block begins to move? (b) The block stops moving just as the extension of the contracting spring reaches zero. Express k (the kinetic coefficient of friction) in terms of s and .