A man pushes a 200N box of books in a ramp by pushing horizontally on it as shown. The coefficent of kinetic friction is 0.30. What force must be exert to make the box move with constant speed? Suppose M is greater than 0.3; show that if exceeds a certain critical value, the man cannot push the box up the ramp. Find the critical value of M.
The force required to make the box move with constant speed is equal to the force of friction, which is equal to the coefficient of kinetic friction multiplied by the normal force, or 200N x 0.30 = 60N.
If M exceeds a certain critical value, the man cannot push the box up the ramp. The critical value of M is equal to the ratio of the force of friction to the normal force, or 0.30.
To determine the force needed to make the box move with constant speed, we need to consider the forces acting on the box.
The forces acting on the box are:
1. The force of gravity acting vertically downwards, which can be calculated as:
F_gravity = mass * gravitational acceleration
F_gravity = 200N (since mass * gravitational acceleration is equivalent to weight)
2. The normal force acting perpendicular to the ramp, which is equal in magnitude and opposite in direction to the force of gravity:
F_normal = F_gravity = 200N
3. The force exerted by the man horizontally, which we need to determine.
4. The force of kinetic friction opposing the motion, given by:
F_friction = coefficient of kinetic friction * F_normal
Since the box is moving with constant speed, we know that the net force is zero. Therefore, the force exerted by the man must equal the force of kinetic friction:
F_man = F_friction
F_man = 0.30 * F_normal
F_man = 0.30 * 200N
F_man = 60N
Therefore, the force that must be exerted by the man to make the box move with constant speed is 60N.
Now, let's determine the critical value of the coefficient of friction (M) beyond which the man cannot push the box up the ramp.
For the box to move up the ramp, the force exerted by the man must overcome the force of gravity acting downwards. The force of gravity can be represented as:
F_gravity = mass * gravitational acceleration
F_gravity = M * F_normal
The maximum force of static friction that can be exerted is given by:
F_static_friction = coefficient of static friction * F_normal
F_static_friction = M * F_normal
For the box to move up the ramp, the force exerted by the man must be greater than the force of static friction. Therefore, we can write the inequality:
F_man > F_static_friction
Substituting the values, we get:
M * F_normal > M * F_normal
Since M > 0.3 (as given in the question), the inequality is always satisfied. Therefore, there is no critical value of M beyond which the man cannot push the box up the ramp.
To find the force required to make the box move with constant speed up the ramp, we need to consider the forces acting on the box. There are two main forces at play: the force applied by the man pushing horizontally, and the force of friction.
First, let's find the force of friction. The force of friction can be calculated using the equation:
Friction force = coefficient of friction * normal force
The normal force is the perpendicular force exerted by the ramp on the box, which is equal to the weight of the box (200N) in this case. Therefore:
Friction force = 0.30 * 200N = 60N
Since the box is moving with constant speed, the applied force must be equal and opposite to the friction force. Therefore, the force applied by the man must be:
Applied force = Friction force = 60N
Now let's move on to the second part of the question, involving the critical value of the coefficient of friction (M). If the coefficient of friction exceeds this critical value, the man will not be able to push the box up the ramp.
To analyze this situation, we need to consider the forces acting on the box when it is pushed up the ramp. These forces include the force applied by the man, the force of gravity acting on the box (its weight), and the force of friction.
For the box to move up the ramp, the force applied by the man must be greater than the frictional force opposing the motion. This can be expressed as:
Applied force > Friction force
The frictional force can be calculated using the equation from earlier: 0.30 * normal force.
Since the normal force is equal to the weight of the box (200N), the frictional force is 0.30 * 200N = 60N.
Therefore, for the box to move up the ramp:
Applied force > 60N
Now, we want to find the critical value of the coefficient of friction (M) where the applied force is just equal to the frictional force. Therefore, we can set up the equation:
M * normal force = 60N
Since the normal force is equal to the weight of the box (200N), the equation becomes:
M * 200N = 60N
Solving for M, we get:
M = 60N / 200N = 0.3
So, the critical value of the coefficient of friction (M) is 0.3. If the coefficient of friction exceeds this value, the man will not be able to push the box up the ramp.