An 8.74-kg block slides with an initial speed of 1.52 m/s down a ramp inclined at an angle of 27.0° with the horizontal. The coefficient of kinetic friction between the block and the ramp is 0.64. Use energy conservation to find the distance the block slides before coming to rest.
The friction force opposing motion is
Ff = M*g*cos27*Uk = 49.4 Newtons
Let X be the distance it travels down the ramp before it stops. When it stops,
(Work done against friction) = (Potential energy loss)+ (Kinetic energy loss)
49.4*X = (1/2)M*(1.52)^2 + M*g*sin27*X
Solve for X
To find the distance the block slides before coming to rest, we can start by using energy conservation. Energy conservation states that the total mechanical energy of a system remains constant if no external forces are acting on it.
In this case, as the block slides down the ramp, there are two forces that act on it: the gravitational force pulling it down the ramp and the force of kinetic friction acting opposite to its motion.
The initial mechanical energy of the block is equal to its kinetic energy, which can be calculated using the formula KE = (1/2)mv^2, where m is the mass of the block and v is its initial speed.
KE = (1/2) * 8.74 kg * (1.52 m/s)^2
= 10.357 J
At the end of the motion, when the block comes to rest, its kinetic energy is zero. The work done by friction is negative and equal to the change in mechanical energy.
The work done by friction can be calculated using the formula W = -f * d * cos(theta), where f is the force of friction, d is the distance traveled, and theta is the angle between the force of friction and the direction of motion.
The force of friction can be calculated using the formula f = u * N, where u is the coefficient of kinetic friction and N is the normal force acting on the block.
N = m * g * cos(theta), where g is the acceleration due to gravity.
Substituting the values:
N = 8.74 kg * 9.8 m/s^2 * cos(27.0°)
= 73.171 N
f = 0.64 * 73.171 N
= 46.837 N
Now, we can calculate the work done by friction:
W = -46.837 N * d * cos(180°)
= -46.837 N * d * (-1)
= 46.837 N * d
Since the work done by friction is negative and equal to the change in mechanical energy, we can write the equation:
46.837 N * d = -10.357 J
Solving for d, the distance traveled:
d = -10.357 J / (-46.837)
= 0.2216 m
Therefore, the block slides a distance of 0.2216 meters before coming to rest.