Physics Questions

Question

1. A 1.2 m long frictionless pendulum of mass 2 kg is pulled 40 degrees from its rest position and let go.  
2. What is the kinetic energy and speed of the pendulum as it reaches the bottom of its motion?
3. What is the kinetic energy and speed of the pendulum when it is halfway to its maximum height?
4. A 120 kg vehicle travelling at 25 m/s slows down to 5 m/s over a distance of 30 m after the driver applies the brakes.
5. How much work was done by the brakes in slowing the vehicle down?
6. How much force was applied to the vehicle by the brakes to slow it down?

Answer 1

Problem 1

a)    In order to find the kinetic energy of the pendulum at equilibrium position, we will find at first its velocity at this point.

We can apply the law of conservation of mechanical energy. Let’s select the zero-energy level - the level at the bottom. Then at point 1, where the pendulum is at angle 40 degrees from vertical, it has only potential energy U = mgh, where h – height against 0s level of energy, m – mass of the pendulum and g = 9.8 m/s² is free fall acceleration. At point 2, at the bottom, the pendulum has only kinetic energy K = ½ mv², here v – the speed of the pendulum at the point.

Now using the law of conservation of energy, we can write

                                     mgh = ½ mv²

Height h could be found from geometric point of view and it is h = l – lcosθ,

where l - length of pendulum, θ = 40° or

                                               h = l(1 – cosθ)

after substitution we have

                                               v = √2gl(1- cosθ)

(Please be careful! Near g is l = length of the pendulum, inside the parentheses just number one – 1)

 Putting all values into the formula, we have     v = √2 x 9.8 m/s² x 1.2 m ( 1 – cos 40°)

Please do the rest of math on your own.

Now put the value of speed v into the formula for kinetic energy K = ½ mv² and you are done.

Part (b) you can do in similar way.

Problem 2

We know that m = 120 kg

v₁ = 25 m/s

v₂ = 5 m/s

d = 30 m

a) to find the work we will use the work-energy theorem: Work done by the force is equal to change in kinetic energy of the body.

W = ΔK

Change in kinetic energy ΔK = K₂ - K₁ = 1/2 m (v₂^{2 -} v₁²). Please find the numeric value. Note that the change in energy is negative. It means that we have dissipation of energy, some of the energy will be converted to internal (heat) energy.

Force could be expressed from the formula for work.

W = Fd.

Bert, I hope now you grasped the problem. Good luck!

Answer 2

1 It is difficult to describe this problem without drawing a diagram, but the approach is as follows. Determine the height above its lowest position that the pendulum bob will have when displaced 40°. Determine the potential energy at that point using the relation mass x gravity x height. This is the value of the kinetic energy at the bottom when it is released. Its speed is then gotten from 1/2 x m x v = KE.

IDo the same for 1/2 height.

2 To do part a, calculate the initial kinetic energy using 1/2 x m x v_i². Then calculate the final kinetic energy using the same formula. The difference in KE is the work done by the brakes.

part b. Use F = ma. The acceleration can be gotten by V_f² = V₀² + 2 x a x distance. Everything is known except the acceleration; solve for it, then use F =ma to find the force.

I got a = - 10 m/s² and F = 120 kg x -10 m/s² = - 1200 Newtons in a direction opposing the vehicles motion.