A box of books weighing 280 N is shoved across the floor of an apartment by a force of 390 N exerted downward at an angle of 35.1° below the horizontal. If the coefficient of kinetic friction between box and floor is 0.57, how long does it take to move the box 4.00 m, starting from rest?
Figuring the pushing force as horizontal and vertical forces is the first key. the horizontal component makes it move, the vertical force adds downward force increasing friction.
frictionforce=mu*forcenormal
= mu*(Weight+290sintheta)
Netforcemoving force=horizontalpushing-friction
= 290*cosTheta-mu(Weight-290sinTheta)
but you know acceleration= netpushingforce/mass
so solve for a.
Then
distance=1/2 a*t^2 solve for t.
To find the time it takes to move the box, we need to break down the forces acting on it and use Newton's second law of motion.
1. Determine the forces acting on the box:
- Weight (force due to gravity), which is equal to the mass of the box multiplied by the acceleration due to gravity (Fg = mg).
- Normal force (Fn), which is the force exerted by the floor on the box perpendicular to the surface.
- Friction force (Ff), which opposes the motion.
2. Calculate the weight of the box:
Weight (Fg) = mass (m) * acceleration due to gravity (g).
Since weight is given as 280 N, we can find the mass of the box:
280 N = m * 9.8 m/s^2
Solving for mass (m) gives us:
m = 280 N / 9.8 m/s^2
3. Determine the normal force:
The normal force (Fn) is perpendicular to the surface and is equal in magnitude but opposite in direction to the vertical component of the applied force (390 N).
Fn = -390 N * cos(35.1°)
4. Calculate the friction force:
The friction force (Ff) is given by the coefficient of kinetic friction (µk) multiplied by the normal force (Fn).
Ff = µk * Fn
5. Determine the net force acting on the box:
The net force (FNet) is the horizontal component of the applied force minus the friction force.
FNet = 390 N * sin(35.1°) - Ff
6. Calculate the acceleration of the box:
Since the net force FNet = mass (m) * acceleration (a), we can use this equation to find the acceleration:
FNet = m * a
a = FNet / m
7. Use kinematic equations to determine the time it takes to move the box:
We have the displacement (4.00 m), initial velocity (0 m/s), and acceleration (a) from the previous step.
We can use the equation d = v0*t + 0.5*a*t^2, where d is displacement, v0 is initial velocity, a is acceleration, and t is time.
Rearrange the equation to solve for time (t):
4.00 m = 0.5*a*t^2
t^2 = (4.00 m) / (0.5*a)
t = sqrt((4.00 m) / (0.5*a))
Plug in the values you found in the previous steps to calculate the time it takes to move the box.