a mass is released from rest from a height of 4m. it slides down a frictionless ramp then along a 20 degree incline section of rough track. how far does it slide along the incline before coming to rest?
friction coeficient= .2
? help!
thanks!
The first thing is to figure the height it fell.
Let the ramp be length L along the sliding service, and d the distance it slid.
Final height h= (L-d)/cos20
height it fell then total is
Heightfell=4-(L-d)/cos20
energy used in stopping is
weight normal*mu*distance
mgcos20*.2*d
so, using energy...
1/2 m(4-(L-d)/cos20)=mgcos20*.2*d
so solve for d. You didn't mention L, the length of the ramp, but you have to have it.
To find how far the mass slides along the incline before coming to rest, we need to use the principles of energy conservation. The initial potential energy of the mass is converted into kinetic energy as it slides down the ramp, and then some of this kinetic energy is lost due to friction along the rough track. When the mass comes to rest, all of its kinetic energy is converted into other forms, such as heat and sound.
1. Calculate the initial potential energy:
The potential energy is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
PE = m × g × h = m × 9.8 × 4 = 39.2m (in Joules)
2. Determine the amount of potential energy converted to kinetic energy:
Since there is no friction on the ramp, the entire potential energy is converted into kinetic energy.
KE = PE = 39.2m (in Joules)
3. Calculate the work done against friction:
The work done against friction can be found using the equation W = μ × m × g × d, where W is the work done, μ is the coefficient of friction, m is the mass, g is the acceleration due to gravity, and d is the distance traveled along the incline before coming to rest.
W = μ × m × g × d = 0.2 × m × 9.8 × d = 1.96md (in Joules)
4. Equate the work done against friction to the kinetic energy:
Since all the kinetic energy is converted into friction, we can set the work done against friction equal to the kinetic energy.
1.96md = 39.2m
Solving for d, we get:
d = 19.9 m
Therefore, the mass slides approximately 19.9 meters along the incline before coming to rest.