A0.23 kg ladle sliding on a horizontal frictionless surface is attached to one end of a horizontal spring (k = 580 n/m) whose other end is fixed. the ladle has a kinetic energy of 170 j as it passes through its equilibrium position (the point at which the spring force is zero). (a) at what rate is the spring doing work on the ladle as the ladle passes through its equilibrium position? (b) at what rate is the spring doing work on the ladle when the spring is compressed 0.66 m and the ladle is moving away from the equilibrium position?

Explanation:

Angular speed of the motion ( SHM )

ω = √k/m

= √(580/.23 )

= 50.20 radian /s

a ) Rate of doing work

= power = force x velocity

At the equilibrium position force becomes zero so

rate of doing work is zero.

b )

If a be the amplitude

1/2 k a² = 170

a = .7655 m

kinetic energy at equilibrium = 1/ 2 m v₀²

1/ 2 m v₀² = 170

.5 x  23  v₀² = 170

v₀ = 3.84 m /s which is the maximum velocity.

Given x = .66 where rate of doing work is to be calculated.

Force at x = ω² x

= 50.20² x .66 =

= 1663.22 N

Velocity v = v₀ √( a² - x² )

= 3.84 √( .7655² - .66 )

= 3.84 x .387

= 1.486 m/s

power = force x velocity

=  1663.22 x 1.486

= 2471.55  W .

trig 15.3 cos

force is based upon mass and acceleration.the statement is not always true.

it can be explained from newton's second law of motion.

as per newton's second law of motion the rate of change of momentum is directly proportional to the applied force and takes place along the direction of force.

mathematically it can be written as -

                                              ⇒      

here   k is the proportionality constant whose value is 1.

                                                            [a is the acceleration=dv/dt]

hence we see that force not only depends on mass and acceleration but also on velocity and rate of change of mass. force f will depend on mass and acceleration only when mass is constant.

hence the statement that force is based upon mass and acceleration is false.