A box of books weighing 229 N is shoved across the floor by a force of 420 N exerted downward at an angle of 35° below the horizontal.If µk between the box and the floor is 0.75, how long does it take to move the box 7 m, starting from rest? (If the box will not move, enter 0.)
To solve this problem, we can use Newton's second law of motion and the concept of friction.
First, let's break down the forces acting on the box:
1. Weight (W): The weight of the box is given as 229 N, which acts vertically downward.
2. Normal force (N): The normal force is the force exerted by the floor on the box in the upward direction to balance the weight of the box. The magnitude of the normal force is equal to the weight of the box, which is 229 N.
3. Applied force (F): The force exerted downward at an angle of 35° below the horizontal is 420 N.
4. Friction force (f): The friction force acts horizontally opposite to the direction of motion and opposes the applied force. The magnitude of the friction force can be calculated using the equation f = µk * N, where µk is the coefficient of kinetic friction and N is the normal force.
To determine if the box will move, we compare the maximum possible friction force with the applied force:
Maximum friction force (fmax) = µk * N
fmax = 0.75 * 229 N = 171.75 N
Since the applied force (420 N) is greater than the maximum friction force (171.75 N), the box will start moving.
Now, we need to calculate the net force acting on the box:
Net force (Fnet) = Applied force - Friction force
Fnet = 420 N - 171.75 N = 248.25 N
Next, we use Newton's second law of motion to find the acceleration (a) of the box:
Fnet = m * a
Since the mass (m) of the box is not given, we can rearrange the equation using the weight of the box:
Fnet = W - f
248.25 N = 229 N - f
Solving for the friction force (f):
f = 229 N - 248.25 N
f = -19.25 N
The negative sign indicates that the friction force is in the opposite direction to the applied force.
Now, we can substitute the value of the friction force into the equation to find the acceleration:
248.25 N = 229 N - (-19.25 N) = 229 N + 19.25 N
248.25 N = 248.25 N
Therefore, the net force is zero, indicating that the box is moving at a constant velocity once it starts moving.
To determine the time taken to move the box 7 m, we can use the equation for distance traveled (d) with constant velocity:
d = v * t
Since the box starts from rest, the initial velocity (v) is 0 m/s. Rearranging the equation:
t = d / v
t = 7 m / 0 m/s
We cannot divide by zero, which means the box will not move. Thus, the time taken to move the box 7 m is 0.